Leland's song is a good guess. But Dana's phrase appears in a number of shanties. "De Boatman Dance" was a minstrel hit, but was it around in 1834?

Leland tells us that he's incorporated some old song into his own poems, but pointedly will not say where. What he does say is, "This is not a folklore book."

Interestingly enough, one of his own poems is called "Time for Us to Go," and the chorus begins with those words. This could mean that he actually heard the shanty mentioned by Dana, maybe at the same time he heard "A Dandy Ship."

OTOH, maybe he just picked up these phrases from Dana's book and built his own lyrics around them. As an American writer of the 19th C., Leland he was surely familiar with "Two Years."

The text of "A Yankee Ship and a Yankee Crew" seems too elaborately poetic to make a good shanty, but the song was obviously very popular and a shantyman might well have used the "Yankee ship" lines with their repeated refrains as the basis for a shanty.

A better candidate for Dana's shanty may be "Tally-I-O!" recorded about 1928 by J. M. Carpenter from James Wright, who went to sea in 1864. Unfortunately that's a full generation after Dana, but the shanty may have been losing popularity for a long time since I can't think of any other texts of it. It is very hard to understand Wright's words - again unfortunately:

                Tally-I-O was a jolly old soul,
                        Tally-I-O! Tally-I-O!
                Tally-I-O was a jolly old soul,
                        Come tally-I-O, you know!        

                What should I do with my rum, Tally-O?
                        Tally-I-O! Tally-I-O!        
                What should I do with my rum, Tally-O?
                        And sing tally-I-O, you know!

                We'll tell [?them we're sober] O Tally-O!
                        Tally-I-O! Tally-I-O!        
                We'll tell [?them we're sober], O Tally-O!
                        And sing tally-I-O, you know!

                Tally-I-O was a [?drunken] old soul,
                        Tally-I-O! Tally-I-O!        
                Tally-I-O was a [?drunken] old soul,
                        And sing tally-I-O, you know!