As Amos points out, the Randolph songs require a different tune from the first two posted and from the Max Hunter song (and from the DT song as well). Like many of these songs, however, it is likely that more than one melody was used.
The songs by Foster and von Smit are, of course, quite different, both in melody and content, but could have furnished the basic idea. Both of them were used on the minstrel circuits and were widespread, reaching the British Isles shortly after their composition and appearing in broadsides.