Although not strictly chanteys, try the following:
"Warlike Seamen" (a fine Copper Family song which traces its origins to the 1600s).
"Captain Ward"
"The Press Gang"
"As We were A-Sailing"
And how can we forget "Heart of Oak" (Burl Ives' verses optional ... for this 1759 classic)?
Not sure, but the following might qualify as 1790s songs too (buy Mystic Seaport's "Link of Chain":
"Paul Jones"
"Captain Kidd."
You might as well just buy Topic Records "Round Cape Horn" for the notes on some of these songs ...
Cheers,
Brian