As well as Dibdin mentioned above, classy composers of the 18th century writing sea ballads (though perhaps without the common touch of Dibdin - one of the first popular singer/songwriters?) also include Arne (Rule Britannia - 1740) and Garrick (Hearts of Oak - 1759).

The (later Royal) Marine Bands were established by 1763 (Portsmouth), 1772 (Chatham) and 1767 (Plymouth) and provided official drummers to the naval ships and fife and drum bands on the larger vessels, though I've no idea of what music they might have played.

(Above info from the National Maritime Museum publication Music of the Sea - David Proctor).

Admiral Benbow died in 1702, so the versions of that may well have been current C18. Admiral Byng (executed 1757) had broadsides written about him. And, with another thread running about him at the moment, Martin Parker just scrapes in being executed for his mutiny of 1797. (Or perhaps these unsuccessful sailors are not quite what you're looking for!)

Mick