The "The Dodson (Dotson) Family of North Farnham Parish, Richmond County, Virginia: A History and Genealogy of Their Descendants Volume One", (The Dodson Book) on pages 27, 28 & 29, has the following statement about John Dodson being the son of Lambeth Dodson. The Dodson Book states and I Quote: "If he [John Dodson] is a descendant of Charles Dodson, Sr. (and the records point to that direction) then his age limits the choice of his parents. His father would almost have to be a son of Charles, Sr. Of the sons of Charles we can probably eliminate Charles, Jr., Thomas, John (because their children were named in their wills) and William (who had no children). This leaves Bartholomew, Richard and Lambeth Dodson as possiblilties. Either choice is reasonable, but we have chosen to include John Dodson in the family of Lambeth Dodson because Lambeth's descendants are known to have settled in the southwestern region of Virginia where John's family is also found. However, the reader should be warned that no information has been uncovered to prove such a connection. It is purely an arbitrary decision on the part of the editor. It is hoped that descendants of this fmaily can use the clues and information offered, and by delving deeper into the records of the counties involved, be able to produce a proved record and lineage." The Dodson Book list the following as known children of John Dodson on page 29: "John Dodson, Jr.; Samuel Dodson, Sr.; and James Dodson".
You list your William as being born 1793 in Pulaski County, Kentucky. Pulaski County, Kentucky was not formed until 1799 from Green and Lincoln counties. Green County, Kentucky was formed in 1793 from Lincoln and Nelson counties. In looking at records you will need to check not only Pulaski county, but also Green, Lincoln and Nelson counties for records on who William's parents might be. Good luck.