My theory is that Don had a wife and family as Dick Whitman before he became Don Draper and married Betty.  He abandoned this family and somehow in his meandering took on the new identity.  He probably didnt become DD right after the war but it was more of a process. A process that included this proxy family.  The dichotomy of the character is always present like when he tells Roger life only moves forward then nxt episode he calls his past. 
This next episode should be interesting to learn more of his past and where he is going?

I think you're right on that

Something about him isn't right

Well, as we learned, Dick became Don during the Korean War. Last episode we found out that he and Betty had been married 9 years (according to Betty), so that would put their marriage in late 1953 or early 1954. So along with selling used cars and taking up with Ann Draper for a while, that would leave precious little time for much else.

Hmm yeah, I also wonder what Don was doing before the war.  I got the impression he didn't see much action by the time he got mistakenly relieved as Don Draper... But if he was 36 in 1961, then he was born in 1925.  So he was 20 by the end of WWII - what was he doing until he got to Korea?  He did volunteer to go, running away from something, as usual... although by 25 you'd think he'd have moved on from his parents.

Donald Draper was a few years older than Pvt. Dick Whitman.  I believe that when Whitman assumed Drapers identity, he also assumed his birthdate and his age. When he was discharged the paper work he he was given had all of the Draper  data.  I don't get the impression that Dick Whitman was 25 years old in 1950/"51 or what ever year he enlisted in the army.  I am sure that somewhere along the series we have been given the clue to Don Whitman's real age.

Maybe I'll play detective and try to track down his true age.

I look forward to  more little tidbits of Dicks life from age 15 until he joined the army.  And the years before his marriage.

I would imagine Dick Whitman joining the army to get away from his "family" as soon as he turned 18. I sure would. And in the flashback, he sure seemed like a scared, untested kid.
Let's say he went to Korea in the first year of the conflict, 1950; that would put his birth year at 1932. So then in 1963, he would be 31. Ok, a mature looking 31.

And a mature-looking 18.