Actually, Lynne, Carter was the son of John Tankersley and his wife Letty (maiden name unknown). His name appears in John's estate records along with the names of his siblings (Fountain, William, Mary, Naomi, Winnie, Rachel and Sarah). All the available evidence is telling me that John was the son of Richard Tankersley, Jr. According to Virginia tax rolls and Revolutionary War records, Richard Jr. had a son named John in exactly the right age group, and this family was living in Henry County, Virginia when your John left Henry County and moved to Georgia.
You'll find records of this family in Greene and Hancock counties, as well as Morgan County. They settled first in Greene, around 1784, and their portion of Greene County became Hancock County in 1793.