John Minear has added some 1830s references. Here (and following) is more commentary.

Here is a reference to the singing of "Highland Laddie" at the capstan in an effort to move the grounded [United States] ship "Peacock" into deeper water on [September] 22nd, 1835, near the Gulf of Mazeira. [coast of Arabia]

It's from an officer's journal. Here's the passage:

"On Tuesday morn, the 22d, the work of lightening was continued, and we saw, with feelings of regret, one half of our guns cast into the sea. The ship was lightened aloft by sending down the upper spars, and unbending the sails; and, on renewing our efforts, we had the pleasure to find that the ship moved and got into rather deeper water. The moment she began to move, new life was infused into all hands, and the men broke forth in a song and chorus, to which they kept time as they marched round the capstan, or hauled in the hawser by hand.
" ' Heave and she must go,' sang one as a leader in a high key, and all the men answered in chorus, in deep, manly tones ' Ho! cheerly.'
" ' Heave, and she will go.'
"'Ho! cheerly.'
" When she moved more easily, those at the capstan, sang to the tune of the ' Highland Laddie,'
"' I wish I were in New York town,'
    Bonny laddie, Highland laddie,' &c.

The shouts of "heave and go" sound like what Dana and others have described. This is exciting, however, in being the earliest (?) such reference to "Highland Laddie." Notably, the lyric uses the "places round the world" lyrical theme that turns out to be of major importance in chanties. This is a really significant piece of evidence! It foreshadows the appearance of "Highland Laddie " as a cotton-screwing chant. It sounds like they started singing Hieland Laddie once the load got lighter and they were able to shift to a march-like tempo.