welcome otdf660 i hope you enjoy your stay here. I have not be to either labadee or falmouth. from what i've read of Labadee it is a private resort island RCCL created to snare more of it's passengers money. Perhaps others here can give more detailed info. as for falmouth i have no info other than you may want to stick to the tourist center in Jamaica, as i have read reports that there have been some criminal acts against people from cruises from time to time. at any rate, be aware of your surroundings and pay attention to the local peoples if you venture out. Grand Cayman is a great beach/shopping day. there are limited attractions/activities at this port unless you dive. the two most prominent attractions are the turtle farm, and stingray city.
http://www.turtle.ky/ couldn't find a link for stingray city, but i did find this
http://www.caymanislands.ky . Seven mile beach is a really nice beach to spend the day if you choose, there are plenty of dining possibilities near/on the beach as well.
cozumel is a nice shopping/dining port. the city center in san miguel is full of shops and restaurants. Peersonally , i rented a car and drove the island myself(kinda hard to get lost on such a small island). There are many resorts and attractions at this port. alot of people seem to like chankanaab park. you can swim with dolphins, stingrays... spend the day at the beach, shop, etc. there are dive shops, mayan ruins of san gervasio, and el caracol. i have always found this a fun port, and have found something different to do each time i've been. if you want to research more about each of your ports of call i would suggest using
http://www.tripadvisor.com it is a very helpful site to research places of interest. there are user reviews which canbe helpful in making your decisions.
I hope you have a great cruise, and come back and share it with us. :camera::beach::cheer: