I believe your William Beacham of Spartanburg Co., SC would be a son of William Beachum (d.1813) and his wife Margaret Lowry. I find the following records for them:
1800 Spartanburg County, SC William Beacham 2-1-0-1-0 // 2-2-0-1-0 // 0-0 1 male 26-44 (ie. 1756-1774) 1 female of the same age 1 male & 2 females 10-15 (1785-90) 2 males & 2 females under 10 (1791-1800)
1810 Spartanburg County, SC William Beachum 0-1-2-0-1 // 0-1-2-0-1 // 0-0 1 male 45+ (1765 or earlier) 1 female of the same age 2 males & 2 females 16-25 (1785-94) 1 male & 1 female 10-15 (1795-1800)
1820 Spartanburg County, SC Wm. Beachum 1-0-0-1-0-0 // 2-0-1-1-0 // 0-1-rest is blank 1 male 16-26 1 female 26-45 1 female 16-26 1 male & 2 females under 10 (I believe this is William the son)
I think the elder William appeared in the 1790 census of Anson County, NC. Household including 2 males over 16, 1 male under 16, 3 females and 1 slave. Next door to him was Susannah Beachum whose household included 1 male over 16 and 2 females. I believe this was his mother, Susannah Hill, widow of Isaac Beachum, and her son Alexander (my 4th great grandfather) and one of the daughters.
Isaac Beachum died in 1777. He apparently owned lands in both Cumberland Co., Va. and Edgecombe Co., NC. Thomas Suggett Hill, of Cumberland Co., Va. was appointed guardian of his minor children. He was, I believe, a brother of Susannah Hill Beachum, they being the children of William Hill and Sarah Suggett.
The minor children were: Darah (Sarah?)Sugatt Beachum, Rebecca Beachum, William Beachum, Jesse Beachum, Daniel Beachum, Alexander Beachum and Betty Beachum. There were at least two older children, not mentioned in the petition, as they were of age. They were: Winifred Beachum (Mrs. John Lowry), and Isaac Beachum, Jr. Doubtless Winnifred Beachum's husband John Lowry was some kin to William's wife Margaret Lowry.