"Meet Me by Moonlight Alone"-- by J. Augustine Wade, c. 1840, provided the chorus to "Meet.../Prisoner's Song" This popular song appeared in several arrangements both in UK and U. S., and in song sheets by De Marsan amd Johnson in the U. S.

The "Prisoner's Song" of the Carters borrowed music from "The Ship That Never Returned." The Carters included a verse 'If I had a ship on the Ocean,...' which may have been a reference to "The Ship TNR."

Who put the two together is debatable. Some claim Newgate prison in London was meant originally, others claim several U. S. prisons. Wade was English.
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Dunno where the murder ballad "The Banks of the Ohio" aka "Banks of the Old Peedee" came from. No 19th c. versions known. TBI notes 1915 as earliest date (where?).

More than one song with that title.

"The Banks of the Ohio" -- Originally "De Banks ob de Ohio," a song of the Nightingale Singers of minstrel fame (F. & W. Stimmel, G. Hunkel, G. Little and J. S. Boswell). Sheet music 1846, but probably a little earlier.
(Also see "Hunt the Buffalo, or the Banks of the Pleasant Ohio")