He was injured and taken straight to the hospital, so no one else that would have recognized him was there.
The real Don Draper did have family - a wife, Anna, who tracked him down and found out another man was using her husband's name. That was Dick's first wife, and he continued to care for her (expenses, house, and friendship) for the rest of her life, even after obtain a divorce to marry Betty.
It is explained that Don grew up without parental units and lived in a brothel raised by various male father figures and whores as stepmother/aunts. His younger brother tracked him down but Don cruelly pushed him away eventually leading to the younger brother committing suicide.
Since Anna covered for Dick/Don, we can assume she told any of the real Don's family that he was KIA. Perhaps while she was trying to track him down, she wasn't sharing with people that he was listed as alive. Maybe she wanted to get answers before telling everyone something was astray... from embarrassment of being abandoned until she had answers? Maybe. She didn't seem like the emotional type to share with people "Don's alive and I can't find him!"
This is assuming Don really had family that would also be asking questions. Since Anna was next of kin, she'd have the real means to find his trail. (Finding out he didn't die in the war). If any other Draper relatives did wonder why he never cane home, maybe in the end, Anna told them Don really did die and it had been an error in claiming he was alive, and they accepted that explanation.
and he paid off the brother who did find him then committed suicide, still think that was the saddest part of the show, Don/Dick telling his brother, probably the one person who loved him unconditionally, to leave him alone.
Even in this day and age if you live off the grid you can't be found. He had the benefit of nothing electronic finding him. Of course you have to suspend disbelief the real Don Draper had no living family except Anna. Who kindly accepted it instead of calling the cops on him. Everyone who knew Don had some dependency on him to keep his secret. The only one that could have turned his world upside down for revenge was the Dr. that he dumped for Megan. That was tied a little too neatly up in a bow. And really it would have been her word against his. And in that day and age Don would have been believed over a woman.