I believe that the book in question (third edition of the original 1955 publication) contains no tunes or information about the songs or their writers/composers beyond the bare texts.   A later edition (Mercier Paperbacks, 1968) contained limited notes by James N. Healy and some of the melodies.  There is also a second volume, edited by Healy (Mercier,1962).  I can provide a midi of the tune (apparantly also called Dangling of the Babies) if you would like.  I don't hear any obvious relation between it and the Dylan/Cash song.

Malcolm