As pointed out by Cox in 1925 there were other early printings that perhaps Deming copied since he didn't copy the Fleet broadside:
"The Gosport Tragedy" was printed in the United States as a chapbook (at Philadelphia?) in 1816, and again (at Philadelphia) in 1829 (Harvard College Library, 25276, 43, 81). It occurs also in The New American Song Book (Philadelphia, 1817), p.69. [Folk-Songs of the South]
I do not have access to these but I can check with the Antiquarian Society- but that's a long shot,