The version in the DT is strictly from camp- scout or whatever. A minstrel song, a spiritual and old doggerel are all mixed in- almost as bad as Alison's. I found one of the spirituals, but not the closest to the minstrel song:
MARY WORE THREE LINKS OF CHAIN (American Songbag, Sandburg). The verse I like best is

My feet got wet in the midnight dew, (3 times)
An' de mornin' star was a witness too;
All my sins been taken away, taken away.

(Who cares about the morning star; my feet are frozen!)

A short version of this spiritual has been posted recently under All My Sins Been Taken Away. It is not the same as "All-er My Sins..." mentioned above (listed separately in the Cleveland Index.
American Memory lists a number of Bland songs, but Hand Me Down... isn't one of them. cufresno says unknown.