US dollars are widely accepted throughout the caribbean so would be better than sterling for when you go ashore. I've not been with princess so dont know if you can withdraw money onboard or not. Most cruise lines do exchange currency but I usually find that you get a better rate getting your currency before you travel in the UK. Sometimes the currency exchange desks are only open for a short time, so they sometimes have long queues to. Hope this helps. Enjoy your cruise - it's a great time of year to be heading out to the Caribbea.
You can take out money from the casino in the evening or at the foreign exchange desk as mentioned above. I usually wait until I get there and take money out as I go along on my switch delta card, so at least I have a smaller chance of loosing it enroute or coming back with loads of money that I did not spend.