This one might be a red herring: Although whaling voyages almost never carried passengers as such so no secondary anecdotal evidence there, one person on board with time on their hands often made anecdotal evidence, the ship's surgeon. Some even wrote relevant poetry.

I have just read a 2003 book which publishes a couple of surgeon journal entries for British whaling voyages in 1831 and 1832, both to Baffin Bay. The following info is more from the author, June Starke, (great granddaughter of the surgeon, George Laing). On page 29 she describes the process of 'tracking' through the ice, i.e., the crew wearing harnesses haul the ship along the edge of the ice.

'If the ice was good they could make about 4 miles an hour as they chanted or sang to the accompaniment of 'tracking pipes'. There is a large body of shanties sung by whalemen to ease such back-breaking tasks. One, the old song "O Logie of Buchan" from the north east coast of Scotland, has a place in George's journal perhaps as his memory of a particular 'tracking' routine.'

This anecdote of the author appears to have been inspired by information from another surgeon's journal as given in a footnote, not Laing's.

'An unnamed surgeon who served on the Hercules of Aberdeen in 1831 writes that during a few days of beautifully serene weather with about 30 sail in company, he enjoyed the experience of ships towing in competition - of hearing the sounds of the men roaring out their shanties, of tracking pipes playing and skippers howling orders from the masthead as they urged their toiling crewmen to greater effort.'
Ref: Journal held at King's College, Aberdeen.

'Lubbock in 'The Arctic Whalers' p56, also records such occasions.'

Whilst I have no doubt about the singing, they certainly didn't in these records use the word 'shanties' and I also doubt very much what they were singing relates to the chanty corpus we know of today. 'Highland Laddie' the possible exception.

BTW the text of Logie O' Buchan whilst very likely from Laing's head is very close to the broadside versions.