They were adopted off-the-books decades ago, scattered by a Georgia doctor who took $100 or $1,000 or something in between to send desperate couples home with new sons and daughters. Now some of the adoptees have turned to fresh DNA testing in hopes of reconnecting with the biological families they never knew, before time runs out.

“This is our shot in the dark, really,” said Melinda Elkins Dawson, one of more than 200 newborns relocated to other states from the clinic in McCaysville, Ga., in the 1950s and ’60s.