Some research links for William Carothers (1785-1826).
There is a record of William Carothers marriage to Sally Carothers in Williamson, Tennessee on Feb 6, 1808. Cousins or married name used instead of maiden name?
Williamson County, Tennessee
County Record Archives: http://williamsoncounty-tn.gov/70/Search-Collections
There is a record of a Carothers Cemetery #1.
Rutherford County, Tennessee