You should definitely go to Dunn's River Falls, but you can save a LOT of money by just hiring your own taxi or minivan just outside the gates of the port area where you dock. We have done this SEVERAL times, the most recent being last March.(I think our ship's tours have been anywhere from about $25 to $35 pp. JUST for the falls excursion.) Also, this way you can go at the time you select. (The earlier, the better.) You will definitely feel like you are in a huge herd if you go w/the ship's excursion. The falls can be crowded enough as it is. (Time of year has a lot to do w/it. We have only been during spring break times.) The ship personnel will tell you what color license plate to watch for to make sure you are getting a sanctioned taxi. Our drivers have always suggested how much time we will need to see the falls, generally l l/2 to 2 hrs. The falls are located in a very clean, beautiful park on the ocean with a beach at the bottom. Our drivers are always waiting for us when we finish, & we never pay until the end of our time w/him. They have also always kept our "stuff" in their cabs & we've never had a problem. (There are also lockers near the bottom of the falls.) We have had them take us back to the ship very briefly, then on to the main downtown shopping street. After about an hour & l/2, they are waiting to take us back to the ship. The more people in your party, the more you will save.(We have paid as little as $6 to as much as $8 pp for the taxi ride, RT, but had to add on about $3 pp when we decided to go shopping, too, as this last time we didn't work it out ahead of time.) Even with the taxi fees & park entrance ($7?? And you must pay it in cash), it's much less than w/the ship. There will be "guides" to "help" you climb the falls or carry your camera, & they get more persistant each year, and require a nice tip, but we never use them. We put our camera in a good plastic bag with a good closure. However, the catamaran ride the other poster mentioned is very nice if money is not a problem. Jamaica is naturally quite beautiful. Be prepared to be hit upon to buy stuff a lot, but if you aren't interested, just say no nicely & keep walking. Even if they take your hand, as long as you are nice but firm, no problem mon! Just keep moving. It's their way and their country. This can be expecially true as you exit the falls park.
Re: Grand Cayman:
Capt. Crosby and Capt. Marvin excursions take you to about 3 great sights to snorkel (if you opt for the l/2 day tour), one of which is Stingray City. However, the ship's tour will only take you to one - but they do have good equipment for you to use, & generally offer at least free punch after you snorkel. We went w/one of the capts. mentioned above on our last trip. (Not the one that plays the guitar.) The tour was fine. The equipment was not that good. No Mae West vests. Just bulky life preservers. And the masks & fins were so-so.
If you have never been to Cozumel, unless you are REALLLY into history, I'd recommend Chankanaab Park (GREAT snorkeling & it's clean - a few bucks for the cab you catch right outside the port area & I think $l0 to get in the park). Or else I'd take an excursion with the ship that takes you on a catamaran ride & to a beach to snorkel. These aren't usually too pricey.
With a family of 6 the shore excursions can really add up! Have a great time!