I assumed it meant they were pliable and moldable.
I can see Randolph, a hunter, saying "doe face," but really, that might describe Randolph himself. Of course one would have to have been around two hundred years ago to be sure, but by all accounts John Randolph had a rather ghostly, spectral appearance himself.
A Doughface or a Copperhead was a "Northern man of Southern principles." I believe there was also a term for a "Southern man of Northern principles," but don't remember what it was. "Scalawag" came along later, of course. Maybe they were just called "unionists."
The term "doughface" lives on. Arthur Schlesinger used it to attack Henry Wallace and the Communist-influenced Progressive Party of the 1940s. Peter Beinhart used it when he wanted to stir up liberal support for the War on Terror. A commentator named David Greenberg trotted it out (unfairly) to attack Eugene McCarthy and (apparently -- it's unclear) to condemn Obama as weak-kneed and insufficiently partisan during the Clinton-Obama primary campaign.
The term originally applied to practical politicians who would always cave in on principle. "Doughfaces" were the RINOs or DINOs of the day. Schlesinger wanted to apply it to liberals or progressives who were too weak stand up to the Communists, but following his usage it came to be applied to head-in-the-clouds idealists who were inept at practical politics, something that doesn't apply well to Fillmore, or Pierce, or Buchanan.
My grandfather, born in 1878 in Texas, used the term "old Texian" to refer to the original settlers. He didn't consider himself to be a Texian because he did not walk, ride a horse or come into Texas in a wagon. That term was reserved (in his mind) for the real pioneers.
John Wayne coined the term “Texican” in a movie about early Texas. I am always a little bit annoyed when I hear someone use the term Texican to refer to the first pioneers.