It's funny that you mention "Round Cape Horn on the Bark Amy Turner," because that was exactly the source I was going to point you towards.
There are a few other books that contain accounts describing songs being learned, but these are almost always accounts of gams or other social situations. I've looked at a lot of books and journals and found that accounts of chanteying are exceedingly rare. You probably know most of these, but here are a few things to check out.
I'm sure you are familiar with R.H. Dana's "Two Years Before the Mast" which has a passage describing how sailors in California picked up the idea of singing at the boat oars from Kanakas.
J. Ross Browne, 1846. Etchings of a Whaling Cruise, contains some interesting accounts of musical activity, including a bit where the crew presses a well-known singer named Tom into giving them a bit of a master class.
William Whitecar's Four Years Aboard the Whaleship. . . , which describes a voyage on the bark Pacific circa 1855, has a lot of accounts of music making, but as is the case with most sailor's writing, he is more inclined to describe occasions of social music making, rather than chanteying.
I would also point you to Frank Bullen's "Songs of Sea Labour (Chanties)", which makes the point made by Radriano above, that the tune was the primary element of the chantey that was learned, and that besides the first verse or two, and the general story line, the majority of chantey lyrics were improvised, or were at least unique to the particular chanteyman. Bullen thought it was improper to print more than a couple of verses of each song because there was never a set text for any chantey, and he didn't want to create a canonical text!
Hope that helps!