White's references to tunes are confusing. For his version A, He mentions the Fisk Quartette "Little David," which was on an old Victrola recording. The chorus is like the modern version (but much extended) and the verse I can't compare with anything I know. I think possibly the quartette made their own tune for the verse.

Listen to an excerpt from the Fisk Quartette here:
Little David

He collected the dance song in the appendix from a book which has no tunes. No idea what was used. One can only indicate the minstrel tunes as a possibility.