As Frank Hamilton noted a while ago, the "Done Laid Around" tune goes back to a couple of gospel numbers, best known of which is "Do Lord."

But if I recall correctly, it is also the tune to WC Handy's Harlem Blues. that could mean that it was written by him or, as likely, that he collected it, as he did for many tunes and songs already in the tradition. The date given in one of the on-line chronologies for The Harlem Blues is 1923.