I should mention that RC Adams was the first to give a musical score. However, he has incorrectly barred the measures, so the rhythm is off/confusing if one is unfamiliar with the song.

After these comes Alden's article, described by Q at the top of this thread.

1882        Alden, W.L. "Sailors' Songs." _Harper's New Monthly Magazine_ (July 1882): 281-6.

Touches of it are unique. It uses the "fishes" theme for the solos. Musical notation is there.

The song then appears in

1883        Luce, Admiral Stephen Bleecker. _Naval Songs._ New York: Wm. A. Pond & Co.

This work reproduces the version from RC Adams, and corrects the faulty notation.