Ms. Lawson searched for the man on Facebook . When she saw his photos, she knew. He looked like their late father. Based on his age and the close physical resemblance, Ms. Lawson immediately told her sister, “He’s got to be our brother.” This was their father’s secret. He had a child they never knew about.

Then came a second shock. Ms. Lawson’s test showed she didn’t appear to have any genetic connection to this new man. This was their mother’s secret: Ms. Lawson was the product of a brief extramarital affair. The man who raised her wasn’t her biological father.

The revelations ricocheted through the family. They created new bonds with people who were once strangers. They caused tension with family they had known all their lives. And they sparked a fight between the sisters about the bonds of loyalty—and how much their parents should have told them.

Ms. Lawson, 65 years old, said she is still grappling with “the pain of knowing my life was a lie and having all these questions that can’t be fully answered because both my parents are gone.”

The hardest part, she said, came the moment she and Ms. Hurwitz, 52, realized they were half, not full, sisters.

“We held each other,” Ms. Lawson said, “and we sobbed.”

At a time of ubiquitous direct-to-consumer genetic testing, family confidences are almost impossible to keep. Companies sell their products for under $100, pitched through heartwarming ads. Millions of DNA kits have been sold in recent years that have handed over information both useful and shocking.

Sales of DNA tests are soaring as people seek to learn more about their roots. Ancestry, whose tests the sisters used, reported sales of 14 million DNA kits world-wide as of November, up from 3 million in 2016. A paper published in Genome Biology, a scientific journal, last year estimated more than 100 million people will have their DNA tested by 2021.

Ancestry provides customers who choose to do so with a way to connect online with others who are DNA matches. The company said it has “a small, dedicated group of highly experienced representatives who speak to customers with more sensitive queries.”

Genetic counselor Brianne Kirkpatrick, founder of Watershed DNA, which provides consultations to people with DNA questions, advises clients to consider how and whether to share certain information. “If you have a secret or think something might be uncovered through DNA testing, start preparing what you want to share ahead of time when you can be in control,” she said.

Ms. Kirkpatrick sometimes suggests clients write letters, explaining details behind their children’s genetic origins, which they can give if the secret is revealed. “I have become of the mind-set it is not a matter of if the secrets will come out,” she said. “It is a matter of when the secrets will come out.”

Given the rapid growth of consumer genetic testing, people can often be identified even if they don’t take a test themselves. Some people who take tests share family trees online. Amateur genealogists and researchers can identify additional connections through obituaries, wedding announcements, and other public information.

In a paper published in October in the journal Science, researchers estimated over 60% of individuals of European descent in the U.S. now have a third cousin or closer relative in a database. “DNA tests can reveal that there is something odd going on,” said Yaniv Erlich, one of the authors and chief science officer of DNA-testing company MyHeritage. “But they don’t tell you the story of what happened.”

In Falls Church, after the test results came back, the sisters sat together and kept staring at pictures of a stranger who looked like their Dad.

Ms. Hurwitz, exhausted and emotional, told her sister she was going to sleep. Ms. Lawson didn’t want to wait another moment. She messaged the man, Dana Dolvin, telling him an Ancestry test showed he was a relative, and suggested they talk. He responded right away. It turned out he lived near Falls Church.

He agreed to meet at Ms. Hurwitz’s home the next day. He assumed they might be cousins.

When they met, the sisters showed him picture after picture of their late father. The resemblance was uncanny, they all agreed; the eyes, the ears, the height, even down to the glasses and love of wearing hats.

Mr. Dolvin, 62, never met the man listed on his birth certificate as his father, who was his late mother’s husband. The couple, both African-American, divorced after his birth. Mr. Dolvin, who has seen pictures, said, “I didn’t look like him.”

When Mr. Dolvin received his own test results, they indicated his DNA was 47% European Jewish. “I kind of figured my Dad was a fair-skinned person,” he said.

He wasn’t sure he would ever identify his father, or even if the man was still alive. Relatives of the man might not want to share private information with a stranger or might not approve of the fact his parents weren’t the same race.

“People don’t want to rock the boat,” he said. “They also may have different feelings toward people of color.”

The sisters say they knew their parents had marital difficulties over the years so it wasn’t a shock to learn their father had an affair. Their parents had a wide circle of friends that included people of different religious and racial backgrounds so they say they weren’t surprised their father had an interracial relationship.

“My surprise was that a child existed,” said Ms. Hurwitz. “And he looked so much like Dad.”

Mr. Dolvin wasn’t certain the women’s assumptions about their father were correct. It was hard to believe, he said, “that I finally got an answer to the question haunting me for such a long time.” He went home and wondered, “Are they really my siblings?”

Siblings share around half their DNA. Half-siblings share a quarter, and first cousins, on average, share 12.5%. Mr. Dolvin checked his report and compared the shared DNA for him and Ms. Hurwitz: It indicated they were half-siblings.

That night, excited by Mr. Dolvin’s visit, Ms. Lawson couldn’t sleep. That is also when she began to wonder why his name hadn’t come up on her results. Why did he have a genetic relationship only with her sister?

Ms. Lawson asked for help answering that question from Larry Alssid, 64, a Long Island, N.Y., psychologist, who had contacted her after he took an Ancestry test a few years ago, showing they were related. They could never figure out their connection but had kept in touch.

After hearing her news, Dr. Alssid suspected Ms. Lawson might have a different biological parent than her sister. He didn’t want to be the one to tell her the potentially shattering information. “I slept on it,” he said.

He told her to check the amount of DNA she and her sister shared in common. Soon she understood: She and her sister didn’t have the same biological parents—or father, to be precise.

“I did regret I told her because she was in shock,” Dr. Alssid said. “We still didn’t know who her father was.”

Later, Dr. Alssid consulted a family tree and gave her names of four brothers—distant relatives he had never met—who he thought could be a match.

Based on their ages, the two youngest, Jack and Ira Greenberg, he said, were the likeliest candidates.

“Jack or Ira? My mother never mentioned those names,” Ms. Lawson recalled saying. She doubted she would find the answers she wanted.

Dr. Alssid told her that a nephew of Jack and Ira might know more. The nephew, whom she emailed, told her all the brothers went by nicknames. The youngest, Ira, was known as Hy.

“That is when I knew,” said Ms. Lawson. “My mother always told me that her first love was a boy named Hy.”

Hy Greenberg was the only brother still alive. An 89-year-old retired traveling salesman, he never married and was living in Florida. His nephew called him about Ms. Lawson’s quest. He agreed to a phone conversation.

She started slowly, telling him she was doing a family tree. Did he know a woman named Brina?

He immediately recognized the name. “Yes,” he said. “I dated her, my best friend introduced us.”

Later in the conversation, she asked the key question. “I have to get personal,” she said. “Would your relationship have included sex?”

Mr. Greenberg said yes.

“You are my father,” Ms. Lawson told him. “I am your daughter.”

“You’ve got to be kidding,” Mr. Greenberg said.

Mr. Greenberg had never done a DNA test. He wasn’t interested. And he struggled to understand how Ms. Lawson could use the DNA test results of other relatives of his to identify him as her father. Later, at Ms. Lawson’s request, he sent in his own kit. The results indicated they were parent and child.

During their first call, he shared details of his early dates with her mother after he got out of the Navy. She wanted to get serious, he said, but he told her he wasn’t interested in marrying. They briefly rekindled the connection years later, after her mother was married. Then they parted ways again.

He never imagined himself being a father, but found they shared a similar sense of humor and a love of storytelling; the conversation lasted hours. She suggested they meet. Mr. Greenberg hesitated, then said, “You want to come, come.”

In June, Ms. Lawson caught a flight to Florida and knocked on the door of her biological father. It was Father’s Day weekend. He called her darling and gave her a hug.

“Welcome home,” he said.

The visit led Ms. Lawson and her sister to have an intense fight. Ms. Lawson posted a picture of herself and Mr. Greenberg on Facebook, and added she was spending her first Father’s Day with her father.

“I was furious,” said her sister, Ms. Hurwitz. “I was in tears. I told her Dad is still Dad, and you have just negated his entire existence and everything he ever did for you with that one post.”

Ms. Lawson said she never meant any disrespect to their late father. “I felt misunderstood,” she said. “My brain was so caught up in what is going on.”

She says she feels a powerful emotional connection to her biological father. The next time she went to see him, she took her sister along.

Meeting him was difficult for Ms. Hurwitz. She kept wondering if he was the man whom her mother preferred over her father. “I didn’t know what to say or how to act.”

Getting to know her new half brother has been easier for her. And it has answered questions for him, too. “I finally got the answer that wasn’t supplied to me by people who loved me and who I loved,” he said.

Growing up, his cousins teased him about the light color of his skin, calling him “white boy,” he said. An only child, he frequently asked his mother about his origins. “Don’t worry about it,” he says she told him. He stopped asking when he was a teenager; his mother died decades ago.

“I was still curious, but no one would tell me,” he said. “Emotionally, you wish it could have been another way, but unfortunately, it isn’t.”

In the months since they met, the sisters and Mr. Dolvin, and members of their families, have met for dinners and outings. During a visit in Boston, Ms. Lawson took Mr. Dolvin around the neighborhood where she grew up, pointing out family landmarks. He refers to both women as his sisters, even though he shares a biological father with only one.

So far, the sisters’ other two siblings, both men, haven’t expressed interest in meeting Mr. Dolvin. Phil Hurwitz, 63, who was born six months before Mr. Dolvin, said he remains unsure “how I want to move forward.”

Ms. Lawson got upset with her brother Phil for not reaching out to Mr. Dolvin. She asked him when he might feel ready.

“I told her I am not putting a time frame on it,” he said. Their other brother didn’t respond to requests for comment.

Mr. Dolvin said he doesn’t think it is his place to contact the two brothers. He said he would let them decide “if they want to welcome me or say ‘hi’…” His voice trailed off for a moment.

“Maybe they don’t feel comfortable with it yet. It’s a lot to take in.”

Ms. Hurwitz said the news that both parents had children from extramarital affairs forced a kind of reckoning she wasn’t sure her brothers were ready to make.

“They have to reconsider completely who their parents were, the lessons they taught us, what they stood for,” she said. “Everyone deals with the emotions differently.”

She looked at her sister, who cried quietly at the table. “It is not up to me to judge the decisions Mom and Dad made,” Ms. Hurwitz said. “It was another world and another time.”

Ms. Lawson said, “I have a hard time when people say it’s the past, move on.”

Mr. Dolvin put his arm around her, comforting her. “I like that we are all together. I’m here. I’m sitting with you.”

There are many unresolved and hard-to-answer questions, such as whether their father was ever told Mr. Dolvin was his son. They don’t think their father knew. “I believe if he knew about Dana, he would have tried to reach out,” Ms. Hurwitz said.

Her father owned a popular kosher deli in a Boston neighborhood. Mr. Dolvin’s mother worked as a cosmetologist in a nearby predominantly African-American neighborhood. Both loved jazz; the siblings speculate the two might have met at one of Boston’s jazz clubs.

The sisters believe their mother knew Ms. Lawson was the product of her own affair. Ms. Lawson and her mother had a difficult relationship, and both sisters think the revelation explains why.

“Julie was a reminder of what Mom did,” Ms. Hurwitz says. “She had to deal with the consequences every day. How did she keep the secret from Dad?”

Both sisters acknowledge they also can’t be sure what either parent shared with the other.

When Ms. Lawson was 29 and Ms. Hurwitz was 16, their parents divorced—and then got remarried nine years later. They stayed together until he died in 2006. His wife died in 2016.

Ms. Lawson says she told her mother she got DNA test results back, but her mother wasn’t interested in talking about them. She died before the second sister took the test whose results revealed so much.

The sisters always return to how much their parents should have told them. Even now, hurt and tensions sometimes flare.

“I understand why you wouldn’t tell,” said Ms. Hurwitz. “The implications of revealing the secret have a domino effect on everyone else in the family.”

Her sister vehemently disagrees. “Every man has a right to know he has offspring,” said Ms. Lawson. “Every child has the right to know her origins. We missed 65 years together.”

Ms. Lawson wears a birthday present she received from Mr. Greenberg, a necklace of two open hearts connected by her birthstone. She is helping plan a party for his 90th birthday in March.

Since the sisters learned the truth, they said they are learning to live with the uncertainties. “I have my anger, my compassion, and my understanding, and I can separate all those emotions,” Ms. Lawson said.

Ms. Hurwitz leaned in closer to her sister. “Every family has secrets,” she said.

—Photo collages created with family photos provided by Julie Lawson, Fredda Hurwitz, and Dana Dolvin.

If you have stories about genetic testing you would like to share, we’d love to hear from you. Please write to Amy Dockser Marcus at amy.marcus@wsj.com