The game runs about 10 times faster as it should and is therefore unplayable. I can't even navigate the menus without having to adjust the cursor several times until it stops where it should.

Apparently Half Genie Hero and Seven Sirens have game speed tied to FPS. Could be that Pirate's Curse does the same thing, try limiting your FPS?

Having enjoyed the Pirate's Curse I went ahead and bought the HGH and it's a pain to play, between it being stuck on something like 30FPS on default settings and after fiddling with driver settings still being limited to 60FPS. Quite unfortunate.

I think it could be two reasons. Setting your monitor to 60hz might help. If that doesn´t work try setting the game to run on 2 cores. It might help since the game is old.

In case anyone else runs into this issue, I solved it by disabling Radeon Chill and Enhanced Sync. I haven't checked so I don't know which one of those two was the culprit, but it works now.