Here's the only snatch of the song I've found of what could be the song in period newspapers; unsurprisingly, it's not bawdy. Actually, I'm not even sure if it's the same song: it seems to be a blend of a couple (at least). As printed Canada's Sasketoon Daily Star on February 22nd, 1922, p. 4 (link)

C.R.E., whose address we are unable to locate on the letter, asks for a poem "The murder of Ben Aulten," of which the following lines are given:
  My name it is Joe Williams,
  My age is twenty-one;
  For the murder of Ben Aulten
  On the scaffold I'll be hung.