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Based on my thirteen years of research into the Chumleys of Spartanburg, there was no person named William Baxter Chumley, just William Chumley. John Chumley (I) (ca. 1740-1750) and wife, Mary, had several children: Thompson/Thomas, William, Dorothy/Dolly, Mary/Polly, John (II), and Sarah/Sallie/Sally.
I believe John Chumley, II married Charlotte Gilbreath, but it is possible that William married Charlotte. However, John Chumley II was the one nicknamed "BUCK." And, many on the Internet say that the one you call "William Baxter Chumley" was called BUCK. Therefore, they concluded the one y'all call William Baxter was the one who married Charlotte. I have some evidence which disputes this claim, and it is part of a twelve page section in my book which gives all the known documentation about the Chumleys of Spartanburg. My book is entitled CHUMLEY-OSHEAL.
There was a Baxter Gilbreath who lived in Spartanburg during the same period of time as Charlotte Gilbreath. I speculate this was her father but have no proof. I believe that is how the name Baxter got into that branch of the Chumley family.