Legman wrote that Joanna Colcord expressed her frustration that she had exactly that problem. She wanted to print what she'd heard at sea in the '90s (or received from correspondents later) but was unable to do so.

My impression is that there must have been a limited number of free-floating bawdy verses, any one of which would instantly produce an "unprintable chantey" by the polite standards of the time - which were notoriously straitlaced anway.

My guess is that most (not all, by any means) of the bawdry appeared in those unrhymed, impromptu verses - including one-off insults directed at the ship's officers and others.