Jeanette - I've done extensive search trying to make a connection to Carter, including his family tree going back to 1540 in Whitchurch, England.
Based on my search the only possible connection to Carter would have been through one of the three of his eldest sons: GEORGE-CORBIN or Carter, Jr.(The other remaining three sons would have been too young). All three of these son's of Carter were born during the 1760 period.
The only connection to Carter and my line of Braxton's would have been by one the three son's; who would have been the father of my GG-GFather Oliver O. Braxton, Sr.
According to the 1850 census of Henry County, Al. Oliver was born 1798. During this period all three of the son's
of Carter would have been in the 30 year old age group; which(Age-wise) would have been possible for either of them to have been the Father of Oliver.
My search indicates that neither of the three had a connection to my GG-GFather Oliver O, Braxton
NOTE: My not being a profesional geanologist I may have missed an important clew, and I challenge anyone to take-up-the-search. William C. Braxton