In TWO YEARS BEFORE THE MAST, Dana mentions the shanty "Time for us to go" ("Leave her, Johnny, leave her") three times, on pages 277, 285, and 301. I think that it is a good candidate for the voyages of the "Julia Ann". Here is an Australian version, from the Carey Collection sung by George Pattison, called "Leave Her Jollies, Leave Her" (Malcolm Forbes, in this same collection, also sings this song):

http://warrenfahey.com/ccarey-s9.htm

From A SHIP OF SOLACE, we have this:

http://www.archive.org/stream/ashipsolace00mordgoog#page/n316/mode/2up

From the C. Fox Smith PermaThread, we have this:

thread.cfm?threadid=85881#1596084

From Cecil Sharp's ENGLISH FOLK-CHANTEYS, we have this:

http://www.archive.org/stream/englishfolkchant00shar#page/2/mode/2up

From John Ward's web site, we have two versions, one from Shay's IRON MEN AND WOODEN SHIPS, and one from David Bone's CAPSTAN BARS:

http://www.jsward.com/shanty/LeaveHerJohnny/index.html

And from Gibb Sahib, an enroute pumping version that seems particularly suited to the last two voyages of the "Julia Ann":

http://www.youtube.com/watch?v=YjNbPMAK7_I&feature=PlayList&p=58B55DD66F22060C&index=130