The only port where I would HIGHLY RECOMMEND a ship's excursion is Falmouth, Jamaica. The reason being, Jamaica is a poor country. If you don't want to do a ship's excursion there, then maybe check out shoretrips.com for excursions that you could book through them. I definitely wouldn't wander off on my own in Jamaica. The other ports are fine. If you don't want to book an excursion in Jamaica, then I would recommend staying onboard the ship.
In Cozumel, I went to Paradise Beach. It's about a $10-15 USD cab ride from the pier. Cabs take US Dollars, so there's no need to exchange your money. There's no entry fee, as the impression is you'll spend money while you're there. Lounge chair use is $2.00, and I think for $12, you can have unlimited use of their water park, floating beach mats, and snorkel equipment.
I've heard good reviews about Chankanaab Park (sorry if I spelled that wrong), which I haven't tried yet, but maybe I will next time to Cozumel.
In Grand Cayman, provided you are able to tender into the port, you can again just get a cab and go to Seven Mile Beach. I've heard good things about that, but again, haven't done it yet. Maybe on the to do list for a future cruise to Grand Cayman. Not to get you down, but if there's bad weather or rough seas, the port of Grand Cayman may deem it too dangerous to run the tenders, and the port will be canceled. This happened to me on my last cruise. I booked one excursion, which was refunded, as well as the port fees for Grand Cayman.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)