I did a similar itinerary back in October of 2010, only we docked in Ocho Rios instead of Falmouth. The only port where I would highly recommend a ship excursion is Falmouth, Jamaica, as that is a port where you don't want to be wandering off on your own in. If you don't do a ship excursion, then maybe find an excursion you like through Shoretrips.com.
It is not necessary to book excursions before boarding, however, some of the more popular excursions sell out fast. One of them is the zipline at Labadee. This is the longest zipline over water, I think at 2600 feet if I recall correctly. In Labadee, you can either wait until you get onboard to book the excursions, or book them in advance. I wouldn't wait until the day of, but that's just me. Labadee is Royal Caribbean's private island, so aside from their excursions that they offer, there are no other excursions. It is a great beach day there if you just want to lay on the beach and watch people go down the zipline.
In Falmouth, do a ship excursion to Dunn's River Falls. This is the "can't miss" attraction in that area. There's several excursions to choose from, so just choose one that you think you might like.
In Grand Cayman, the "don't miss" attraction is Stingray City. Having only been there once, I can't recommend a certain tour vendor. If you haven't been there before, then I might recommend booking through the ship.
In Cozumel, that really depends on what you want to do. Cozumel is safe to do on your own, and not book a ship excursion. My last trip there, I just took a cab to Paradise Beach. I've heard Chankanaab (?) is nice also, but haven't been there. Maybe that will be on my list for the next time I go to Cozumel. Cabs are easy to get right after you exit the pier area, and if you go to a beach club, they are also lined up and waiting to take you back to the ship.
If you haven't been to a port before, then I would recommend sticking with the ship's excursions. Also bear in mind that if you book through the ship, if for whatever reason the excursion is late getting back, the ship will wait for you. However, if you go out on your own, the ship WILL NOT WAIT for you. Make sure you keep your watch on SHIP TIME, and not ISLAND TIME. It's at the Captain's discretion whether he decides to change the time or not.
Enjoy the Freedom! Like I said, I was on it in October, and my review of that is here.