On the CD/Record Blow Boys Blow there is a version of this song By (Now I might be wrong but is it not) A.L. Loyd (not Bert? Or did his friends call him Bert? ) Who was the shanty man in Moby Dick. The other performer on this record was Ewan MacColl.The verses are the ones that Bat Godess posted, whithout the one about the **** and in a different order.
The most interesting part for me is the wild yelping "To me way-ay-ay EAUGHA!" (Oh so that is how you spell it!) that comes at the end line.
As for Moby Dick I do remember that I heard the shanty once in the movie, I dont know about there being two other versions in the film but I may have missed them. I did recognise some other shanties in the film but the names escape me at the moment.
Here is a different set of lyrics I found at this site http://argo.acronet.net/~robokopp/shanty.html
To my, Aye, And we'll furl,Aye,And pay Paddy Doyle For his boots.
2. We'll sing,Aye,And we'll heave,Aye,And we'll hang Paddy Doyle For his boots.
3. We'll heave, Aye,With a swing,Aye, And we'll all drink Brandy and gin.
So as is often the case it is hard to say what the "real" lyrics are.