There is evidence that "Drunken Sailor" goes back considerably further than we thought.

This site:

http://www.pdmusic.org/1800s.html

includes a MIDI for the composition "Drunken Sailor (Rondo/Divertimento)" by Christopher Meineke (1782-1850). The piece is said to have been published in 1825.

Its first strain is the major version of the shanty tune (essentially "One Little Two Little Three Little Indians"). At least one shanty authority (can't remember which one) states that this was a "modern" variant.

Either way, Meineke's composition suggests that the shanty, with one tune or the other, was in use long before the Civil War.