So far as I know, only one traditional tune for this song has been reported, once by Colcord and once by Harlow.

Harlow may have cribbed the tune from Colcord, but his text is longer.

Harlow, from a sea-faring family, sailed on (I believe) three voyages in the 1870s.

After thinking about it a little more, I now believe it most likely that Colcord's "offshore whaling" simply means "off the West Coast of South America" - as in the song!

Just possibly an early reference to the song appears in Henry Mercier, "Life in a Man-of-War" (1841):

“Dobbs...had been for twenty or thirty minutes desperately engaged in drawling forth a song to which there appeared to be no end, the burden of which was the capture of a whale after a severe and bloody struggle....

“‘Why, damme, man, whilst you have been lowering the boats away in that song of yours, a good Nantucketer would have three whale turned up.’
“‘Oh !…The marrow of the ditty is to come yet—when we get the fish alongside.’”

(Mercier also gives an unmistakable fragment of "The Greenland Whale.")

Captain Harry Allen Chippendale gives three stanzas in his autobiographical "Sails and Whales" (1951). He signed on a whaler in 1895 at the age of sixteen:

"Our waist boat got down and of course got the start,
'Lay me on, Mr. Howland, I'm hell for a dart.'
Now bend to your oars and make the boat fly,
But one thing be sure of, keep clear of his eye!

"Now Dukey is fast, and the whale has gone down,
And the first mate lies waiting his line to bend on.
Now the whale has come up, like a log he does lay;
Whatever he done, boys, he gave us fair play.

"But we fought him 'longside, and a lance we did thrust,
Then he rolled and he rolled, and he flurried and fussed,
Which caused him to vomic and blood for to spout;
In less than ten minutes, we had him fin out."