Jon has already suggested some likely candidates, Benbow, Captain Kidd and Jack Hall all dating to about 1702 or soon after and all to roughly the same tune. 'The Barley Mow' was around by then but very likely not in the form we sing it today. What could be the original is in one of Ravenscroft's books of 1609.

Some probables

Child 285 'The George Aloe and The Sweepstake' from the 17th century, likewise 'Sir Walter Raleigh sailing in the Lowlands' or as we know it today 'The Golden Vanity' Child 286.

Child 287 'Captain Ward and the Rainbow'.
Versions of all of these are still sung today so finding tunes shouldn't be difficult.