I haven't time to write out my notes to the second edition of Wanton Seed just now, but a close look at the 1805 broadside throws up some interesting detail. It was being performed in the theatre by several well-known actors of the day as 'A Whimsical ballad'. I translate this to mean it was probably being burlesqued as were many traditional ballads during the 18thc. The ballad has several times been rewritten/updated/relocated to include details about more current events and therefore appeal more to more recent audiences.
See 'Handsome Harry' (Laws P34B)
'A New Song on the Ghost of Mary Ashfield' etc printed in Birmingham, in the Madden Collection.

The likelihood is that it was printed on one of those 3-4 column broadsheets c1750 by the likes of the Diceys and took off from there.

Longshot...could have been a parody on one of the other 'Jonah' ballads, Demon Lover, Captain Grant, Cruel Ship's Carpenter etc.

Can't help with tunes I'm afraid.