That does it for the references I have, at present, to "shanty sightings" attributed to the 1820s and a few decades earlier. More could turn up if one searches creatively with different criteria. I realize it is not a lot to go on, but I would be interested, at this point, to try to think about the scene only based on the available evidence (there must be a bit more I am overlooking?), and try to block out our later impressions. Pretend the 1830s have not happened yet. What do we have?

1780s-90s:

General references to African and New World Black work-songs, from Mali, Grenada.

1800s:

General references to African-American work-songs and their style, from Martinique;
Rowing songs from Georgia, South Carolina, Guyana, Surinam;
Windlass songs, aboard vessels with sailors incl. from Northumberland and Holland.

1810s:

2 stevedore songs from Jamaica that resemble chanteys;
African-American rowing songs from Antigua, Virgin Islands;
Singing and fife-playing at the capstan on a British war ship.

1820s:

Rowing songs, from Georgia, Virginia, St. Thomas;
A version of "Cheerly Men" for topsail halyards on a brig near Quebec;
Fictional capstan shantying in the Arctic; capstan (?) song of British tars in London; chant for pulling known to an ex-British navy man. [I've also seen another reference to the phrase "British capstan song" from 1825.]

Anything more, strictly from these time periods?
Analysis to come later! ;)