If you're going to "remove anachronisms" from well-known shanties, you'd probably be better of just writing new songs.

Because the entire shanty would be an anachronism.

But the larger point is that sailors of ca1700 *didn't have* shanties. I seem to recall a mention of merchant seamen in the later 18th century weighing anchor while singing "The British Grenadiers," a well-known shore song.

That case seems to have been exceptional. But since heaving a capstan is slow, steady work, it would certainly lend itself to some kind of singing - even if shanties, strictly speaking, were as yet unknown.

Or by "shanty" do you (like so many other people) just mean a "song related to the sea"?