In the early hours of 16 August 1977 Elvis played a short game of Racquetball then sat at his piano and sang a few songs, including 'Blue Eyes Crying In The Rain'. That's most likely the last song he ever sang: at 1.30pm he was found on the bathroom floor and his death was officially recorded at 3.30pm.
As for studio recordings: Elvis recorded several new songs in the "Jungle Room" at Graceland on 29-31 October 1976. The songs recorded were It's Easy for You, Way Down, Pledging My Love (all on the 29th) and on the 31st - ironically? - the last studio track: He'll Have to Go.
The last official live recording was at the Rushmore Plaza Civic Center in Rapid City, South Dakota on 21 June 1977 (tracks appear on the LP Elvis In Concert and the CBS TV special of the same name). And his final concert was aat the Market Sqaure Arena in Indianapolis on 26 June 1977.