Hi, I am also tracing this line. My line branches off of Thomas R. Pulliam and his son Judson Pulliam. I do not have a lot to add at this time, I believe Thomas's father was Richard from other research and reading I have done, but I have just started tracing this line and have not pulled it together yet. I would like to stay in contact and let you know of anything I find.
Oh, Judson Pulliam was my Great Grandfather, which makes Thomas R. my GG Grandfather and of course your William Thomas my GGG Grandfather.