This next reference from the 1870s is intriguing because it seems to relate to Adam's ON BOARD THE ROCKET. The text is A VOYAGE AROUND THE WORLD (1871), by Nehemiah Adams.

If I am getting it right --the text may need further examination-- Nehemiah was the father of R.C. Adams of "Rocket" fame and his 1879 account. The younger Adams' adventures seem to have been in the late 1860s. The voyage that Nehemiah describes looks to have been started in October 1869, *after* the stuff that happened in ON BOARD THE ROCKET. The father went to sea for his health, aboard a ship GOLDEN FLEECE, of which his son was the captain, and which was bound out of Boston for Frisco, Hong Kong, and Manila.

First he mentions some pump shanties.

...the boatswain's "Pumpship " at evening, when twelve or fifteen men entertain you with a song. Every tune at the pumps must have a chorus. The sentiment in the song is the least important feature of it, — the celebration of some portion of the earth or seas, other than here and now : "I wish I was in Mobile Bay, " " I'm bound for the Rio Grande," with the astounding chorus from twenty-eight men, part of whom the fine moonlight and the song tempt from their bunks, is an antidote to monotony.

The first named probably refers to "Knock a Man Down". But neither that (BLOW THE MAN) nor RIO GRANDE are usually associated with pumping, so...

The sailors were a merry set. Though only half of the crew—that is, one watch—were required each night at the pumps, all hands at first generally turned out because it was the time for a song. It was a nightly pleasure to be on the upper deck when the pumps were manned, and to hear twenty men sing. When making sail after a gale, the crew are ready for the loudest singing, unless it be at the pumps. For example, when hauling on the topsail halyards, they may have this song, the shanty man, as they call him, solo singer, beginning with a wailing strain:

Solo : O poor Reuben Ranzo! (twice. [EACH LINE IS REPEATED])
   Chorus: Ranzo, boys, Ranzo!
Solo: Ranzo was no sailor!
Chorus: Ranzo, boys, Ranzo!
Solo : He shipped on board a whaler!
Chorus: Ranzo, boys, Ranzo!
Solo: The captain was a bad man!
Chorus: Ranzo, boys, Ranzo!
Solo : He put him in the rigging!
Chorus: Ranzo, boys, Ranzo !
Solo: He gave him six-and-thirty —

by which time the topsail is mast-headed, and the mate cries, "Belay!"

When the mainsail is to be set, and they are hauling down the main tack, this, perhaps, is the song : —

Solo: " 'Way! haul away! my rosey ;
Chorus: 'Way ! haul away! haul away! JOE!"

the long pull, the strong pull, the pull altogether being given at the word "Joe;" then no more pulling till the same word recurs.

When hauling on the main sheet, this is often the song, sung responsively : —

Shanty man: "Haul the bowline; Kitty is my darling.
       Crew: Haul the bowline, the bowline haul!"

Now, these are some of the exact same shanties --same lyrics-- as RC Adams published in 1869. What to make of that? Why did the son, the experienced sea captain, need to reproduce the exact shanties as his dad, a passenger? Did he dictate these lyrics to his father in 1871? Minimally, if the father did hear these in 1869 (and out of some laziness, the son reproduced the same), then we need to date RC Adam's ROCKET shanties to 1869 (a minor change in dating).

FWIW, this Adams also uses the term "shanty man," but not the term "shanty."

The text is here.