I've never heard of "Mary Anne" used as a shanty, but almost anything is possible.

The most likely shanties to have been sung are those (mostly too familiar today) known to have existed in the 1850s. Many good ones, including some favorites, could have originated long after the voyage of the "Julia Ann."

My guess too is that the lyrics actually sung during the voyage were shorter and varied from the texts in most collections. That's simply because of the nature of shantying.