From Dana's list of shanties sung on board the "Pilgrim" and "Alert" between 1834 and 1836, published in TWO YEARS BEFORE THE MAST (page 285) I would suggest "Cheerily, Men". Here are several early versions of this song. First of all from Warren Fahey's site in Australia:

http://warrenfahey.com/Sydney-Folklore/SECTION-7/sfp-7-Cheerily.html

And then from Laura Smith's THE MUSIC OF THE WATERS, p.22-23:

http://www.archive.org/stream/musicofwaterscol00smituoft#page/n59/mode/2up

From the C. Fox Smith Perma Thread:

thread.cfm?threadid=85881#1595202

From an old book called THE SHIP OF SOLACE:

http://www.archive.org/stream/ashipsolace00mordgoog#page/n252/mode/2up

And from Cecil Sharp's ENGLISH FOLK-CHANTEYS :

http://www.archive.org/stream/englishfolkchant00shar#page/50/mode/2up/search/Cheerily+Man

And here is Gibb Sahib's EXPLICIT! version on YouTube:

http://www.youtube.com/watch?v=o9xoIfGQwZs