Not really in the caribbean it is better to get off the ship and if you have another couple that you can share with - hire a cab and see and do what you want. If you want to do some diving then it is worth the piece of mind to pay the extra on go on a shore excursio.
It is better to do your own thing in the Caribean, unless you want to see a specific site. We normally just get a local taxi/bus and off we go. The islands are similar and if you want the beach it is cheaper to use the local transport who will take you and agree a pick up time, and if a taxi will detour to show you more of the island. Do go to Megans Bay in St. Thomas the journey itself is an experience and the beach outstanding. We are doing is again in January and doing our trips on our awn. If you go to Cozumel there is a local transport/taxi which takes you to a leisure park and for ten dollars each day you can have a great day sun bathing and swimming with the dolphins (extra) but worthwhile doing and get there early as there are normally quite a few ships in Cozumel each day/ Have agreat time .
If going to jamaica, do go on an organised tour - it really isn't safe to do your own thing there though this is the only island we found this to be the case. On the other islands it depends if you want to go somewhere specific or do something organized e.g. sea kayaking etc. Usually it is cheaper to get off the ship and you will find yourself surrounded at the harbourside by local taxi drivers who are really friendly and willing to do your own private tour for half the price of Princess. Just agree the price and check they've got air conditioning before you agree but they are usually very good. It's cheaper to join up with another similar minded couple and split the cost. Hope you have a great time.