Will, it appears that Thomas P. Hamblen was probably born in Franklin County, VA. I have seen documentation that Pierce Dant Hamblen, his father, was in Franklin County VA in 1793. Franklin County is next to Pittsylvania County, VA, where Pierce Dant resided at least through 1787.
The bad news is that I cannot find that documentation now, but I keep searching. There is no info that I know of that Thomas was born in Sullivan County. The documentation I am looking for listed Pierce Dant as Dant Hamlin, which was common for him. He evidently went by the name Dant.
Also, the Mullins connection does not exist. The children of Swift and Caroline were not part of Thomas' probate. The probate book had a formatting error.