Okay, I've managed to get time to start looking at some of my references. One interesting and tantalising reference I have comes from the Times Aug 15th 2008. It refers to a sailor's journal up for auction, George Hodge, and covers the period 1790 to 1833. Apparently he was in the RN from 1790 to 1815 then joined the MN. Quote 'It includes everything from the lyrics of sea shanties to a picture of the first ship on which he served.' The auction was at Northeast Auctions in Portsmouth, New Hampshire and the whole collection of his memorabilia was estimated to fetch about $50,000. I didn't manage to find out anything further, but Googling might bring up something. It would be great if the journal had been published???? or was on its way to being published.
Anybody else spot this one or know what happened to the journal?

I must add that journalists and those with a passing interest often mistake sea songs for shanties, so it may be a red herring. Even if it is it would be interesting to see what the sea songs were.