Thank you very much for all of your help. I don't know the broader significance of the name Pierce in the Dunham family, but George's son Pierce was my great-grandfather. His first name is Edward and he seems to have alternated between use of both names. Pierce had three sons, Elmo Preston Dunham (who is my grandfather), George Wallace Dunham, and J. P. Dunham (who died in infancy). My family does not know what "J. P." stands for, but it seems reasonable that the "P" could stand for either Pierce or Preston. All three sons are buried in Vidalia, Georgia, which is my hometown.

I've also found an Elisha Dunham (b. 1774, CT; d. 1884) who married a Jerusha Clark (also listed as b. 1774, CT; d.1844). I haven't been able to link them to the other Connecticut Dunhams and I can't find a record of them having had children. The 1810 Connecticut census only has 2 Elisha Dunhams listed as heads of households. However, neither of those two households includes a man aged 26-44, so neither seems to be this Elisha (who might not've been the head of his household, or who might've lived outside of Connecticut by 1810, or who might've used another named for the census, or whatnot).

Here's the information I have on the two Elisha Dunhams listed in the 1810 census:
W, Hartford: 3 males under 10, 1 male between 10-15, 1 male between 16-25, 1 male 45+, 1 female 10-15, 2 females 16-25, 1 female 26-44 ===the problem with this one is that Elisha would’ve only been 36 years old at this time, which isn’t reflected in the census.

Mandsfield, Windham: no males under 10, 1 male 10-15, 1 male 16-25, 1 male 45+, 2 females 16-25, 1 female 45+===same problem as above.

Of course, I don't even know that the Dunham I'm looking for is named Elisha or Elijah, I only know that his son is. Nonetheless, the household described above could possibly be the childhood home of my great-great-great grandfather, as there were three males under 10 living there when he would've been 6 years old (or 8 years old, depending on which record you go with).

There was a different Elijah that moved from Connecticut to the South around the same time as my great-great-great grandfather, but he was older at the time and went to North Carolina rather than Georgia. He appears in some academic articles about peddling in the American South during the early 19th century.