Stephen Foster wrote a "Plantation Melody" called "Oh, Boys Carry Me Along!" in 1851, the words of whose chorus may have contributed to the sea shanty.
(Lyrics here: http://www.pdmusic.org/foster/scf51i.txt).

Foster also wrote "Santa Anna's Retreat from Buena Vista" in 1848 - one of his less satisfying tunes, I'd say. (MIDIs for both songs here: http://www.pdmusic.org/foster.html) I don't hear any resemblance to the old-time tune "Santa Anna's Retreat."

Cecil Sharp collected a version of the shanty before 1914. "Revival" versions are easily identifiable by the modal dip on the penultimate note of the chorus. Steeleye recorded it around 1972; don't know if they were responsible for the variation.