Elvis Presley recorded "Always On My Mind" (Christopher - James - Carson) in 1972 as the flipside of "Separate Ways" shortly after his divorce. He reached the top twenty. It's on the album "Separate Ways". What else would you issue after a public divorce?
First to record "Always On My Mind" six months before The King was Brenda Lee as the final track on the album "Brenda" in 1971 but other than showing on the Country chart, the original went nowhere.
Willie Nelson won a Grammy for his top five cover in 1982. The composers also won a Grammy for Song of the Year. You can find it on the album "Always On My Mind" which won Album of the Year.