Unfortunately, finding period shanties is not as easy as it seems. There are lots of sea shanties but most of them are from the 19th century. You don't even have to research them. The subject matter is a either about whaling or shipping. Sometimes historic events or people that post-date the GAoP are mentioned.
For example, any song that mentions Australia is too late. It wasn't settled until 1788. South Australia was founded in 1836. This eliminates the song South Australia, also known as the Rolling Kings.
Haul Away Joe starts with a verse about King Louis XVI who was executed in 1793.
Donkey Riding refers to a steam winch used to load timber onto ships.
The Bonnie Ship the Diamond is a whaler.
It goes on like this.
I've heard but can't confirm that the sea shanty as we know it wasn't really common until the 19th century. Prior to that musicians, especially fiddlers, kept time.
We've eliminated sea shanties as period music. What's left for pirates? There are some songs with sea themes. Also, there are two that I know of the specifically mention pirates. The first one was written by a young Ben Franklin about Blackbeard's death. It is called the Downfall of Piracy. Unfortunately only the words survive.
The other one is My Name is Captain Kidd. This one includes music and you can find versions of it on YouTube.
