I can't help but mention one of my favourite songs - The 51st Highland Division's Farewell to Sicily:

http://www.scottishpoetrylibrary.org.uk/poetry/poems/51st-highland-divisions-farewell-sicily

Now, this doesn't really count. It was written by Hamish Henderson, not a sailor but a soldier in WW2, and set to a traditional tune. With verses like

Sae fare weel, ye dives o’ Sicily
(Fare ye weel, ye shieling an’ ha’),
We’ll a’ mind shebeens and bothies
   Whaur kind signorinas were cheerie.

It draws so strongly on the traditions of sea songs that I think it's fair to bring it up in this thread. It shares a lot in terms of mood and attitude with the kind of songs you're looking for, even if it doesn't fall strictly within the "written by a sailor" limits.