I'm afraid I'm not musically literate enough to have an opinion on what is the true and correct tune for ALL AROUND MY HAT (I normally deal only with lyrics)--but this might be useful to somebody:

A musical arrangement of ALL ROUND [sic] MY HAT, with 3 verses, appears in The Franklin Square Song Collection, Volume 8 (New York: Harper & Brothers, 1892), page 125.

I haven't compared the tune or lyrics to what has already been posted.