In the DT:

BLACK JOKE  Text of c.1730, with tune and extensive notes.
MY GIRL SHE'S AIRY  Text from The Merry Muses of Caledonia, noted and perhaps modified by Robert Burns.
BLACK JOKE (3)  ("The cole Black Jack", c 1745)
BLACK JOKE (4)  ("The whore's Answer to the Rakes")

BLACK, WHITE, YELLOW AND GREEN  Text only.

Much of the above material came from Bruce Olson; that and more can be seen at his website:

The Black Joke  A comprehensive history of the song.

There are a number of other songs at the Bodleian which were set to the tune.

Malcolm