Steve, I've looked at 3 printings of "Molly the Betrayed" at the Bodleian site, all pretty much the same apart from a few words here and there and a few spelling variations. They have the lines:

That night as asleep in his hammock he lay,
He fancied he heard some sperrit for to say,
'Oh, vake up young Villiam and listen to hear,
The woice of your Molly vot lov'd you so dear.

Your ship bound from Portsmouth, it never shall go,
'Till I am reweng'd for my sad overthrow, [...]

which are similar to:

[...] But as in his cabbin one night he did lie,
The voice of his sweetheart he heard to cry.

O perjur'd villain, awake now and hear,
The voice of your love, that lov'd you so dear;
This ship out of Portsmouth never shall go,
Till I am revenged for this overthrow.

Those two stanzas are in the Roxburghe-type text, but not in "The Cruel Ship's Carpenter" (or in the Deming version, for that matter).

That would be my main reason for thinking the burlesque must have started from the old broadside version, rather than from CCS, but it also has new material that doesn't have a close parallel there or in CCS.