I've yet to visit/re-visit shipboard worksong references for the 1840s.

Olmsted's INCIDENTS OF A WHALING VOYAGE (1841).

Olmsted was in the whaler "North America," sailing out of New London (CT) to Tahiti and Hawai'i. Here's what he says, for 1840:

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Tuesday. Feb. 11. I have often been very much amused by the cries and songs of the men, when engaged in hauling away upon the rigging of the ship. The usual cry is " Ho ! Ho ! Hoi !' or " Ho ! Ho ! Heavo !" which is sung by some one of them, while the rest keep time. It has a rather dolorous cadence, and a wildness that sounds like a note of distress when rising above the roar of the gale at dead of night....

[Sounds like "singing-out"]

...But there are many songs in common use among seamen, of a very lively character, which though bereft of all sentiment and sense in many instances, are performed with very good effect when there is a long line of men hauling together. Mr. Freeman usually officiates as chorister, and with numerous demisemiquavers, strikes up the song, while all the rest join in the chorus. Sometimes they all sing together as I have endeavored to represent, although it must appear very tame without the attendant circumstances. One of the songs is as follows:—

Ho ! Ho ! and up she ris - es
Ho ! Ho ! and up she ris - es
Ho! Ho ! and up she ris - es,
Ear-ly in the morn-ing.
[with music, tune shape is "Drunken Sailor"]

And another song, accompanied with the chorus, which vies with the song of the troubadours in poetic sentiment.

Nan. cy Fan - an - a, she mar - ried a bar - ber,
CH: Heave her a - way, and heave her away
Hurrah, hurrah, for Fancy Fa-na - na.
CH: Heave her a - way ! and Heave her a - way !
[with music, in 6/8 and very similar in shape to Hugill's "Haul 'er Away," though the tune is different, it seems to be a variation.]

There are many other songs that might be very easily mentioned, which, however, like a good proportion of our parlor songs are rather insipid without the music. The songs of sailors, when sung with spirit and to the full extent of their fine sonorous voices, add new vigor to their exertions, as the heavy yards and sails are mounting upwards....

[On a later occasion:]

...The teeth of the sperm whale vary from four to five inches in length, and are imbedded more than two-thirds in the lower jaw. They are susceptible of a very high polish, and are beginning to be valued as an article of merchandize, which has induced sperm whalers to collect all the teeth of their captured whales, as constituting a part of the profits of the voyage. The extraction of the teeth is the practice of dentistry on a grand scale. The patient, i. e. the lower jaw, is bound down to ring bolts in the deck. The dentist, a boatsteerer, with several assistants, first makes a vigorous use of his gum lancet, to wit, a cutting spade wielded in both hands. A start is given id the teeth, while his assistants apply the instrument of extraction to one end of the row, consisting of a powerful purchase of two fold pulleys, and at the tune of "O! hurrah my hearties O!" the teeth snap from their sockets in quick succession.

[The pulley arrangement rigged appears in an illustration. The men are hauling in a downward direction.]

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These chanties look like they are probably the "older" ones. If we are to assume "Drunken Sailor" one was a walk-away, then one could imagine that as a continuation of that practice since, for example, Navy days. "Nancy Fanana" smacks of "Cheerly Men." It does have a possible "stanzaic" structure, that would lend itself to the "double pull" halyard maneuver. On the other hand, there is no description of that, and it is possible that it was timed like Cheerly Men. On the other hand, the command is "heave," and it is not explicitly connected to halliards, so it could have been for another task. I am really not sure.

The song for pulling whale teeth, "O! hurrah my hearties O!," to which we've seen songs with similar generic phrases before, looks like it may have had the form of a sing-out or of a Cheerly Man type deal, i.e. with the pull on the last "O!" I say that because of the nature of the action being described, which involves a stiff downward pull (e.g. as in sweating up).