This topic really should be in the Caribbean Port chat section due to the ports you are visiting but since you are new to the site I won't confuse you by moving it.
Nassau- I can't offer much more than I have already for Nassau. You certainly don't need a ships tour here. Maybe someone that has been there lately can jump in. I think Dwayne is coming back from a cruise that went there so maybe he can provide up to date info.
Cozumel- Very easy to do on your own. Walk out of the port shopping area. You will come to an area where all the taxis are. You will find a sign that lists prices for various beach clubs. With as many people as you have you might be able to negotiate a better price and you should try. But you should be able to find a beach club that has showers/bathrooms/food and bevs available. The taxi/tour guide will probably stay with you the entire time and drive you back when you are ready. Even if they leave you at the beach club you can easily get a taxi back from anywhere they might take you. Just make sure you keep your watch on ship's time and plan to be back an hour before sailing and you should have enough buffer time in case you run into a delay. DO NOT try to go to the mainland and see Tulum on your own.
Roatan- We love Roatan. Great beaches and snorkeling. Last time we were there (June) we used Roatan Taxi Tours. If you Google on Roatan you will find them. They can give you a tour of the island and stop at a great beach if you ask. You will save at least half of what a ship's tour would run you. We go to Foster's West Bay. for a beach. They have bathrooms, food and a bar available. But ask you tour guide where else you may go. We also did zip lining on Roatan. Very reasonable. Priced about a 1/3rd of the ships tour (per person)
Belize- Belize is an oddity. We like Belize but there really isn't anything to do in town.
They have jungle tours, cave tubing (which is fun unless you are claustrophobic) zip lining etc. They also have great snorkling but you have to motor boat out to the barrier reef. If you want to snorkling/neach break I suggest you book through the ship. It isn't that much more expensive and they pick you up form the ship so you have more time and its easier.
If you go into port, you have to venture out into the country side to see anything it's famous for. BUT if you google on tours and Belize you will find options on doing this port on your own. You can even just take the tender into the port and there will be tour operators there that can sell you a tour on the spot. You can probably get a better price if you book ahead though. Don't be afraid to do this as every tour company I have dealt with will refund your deposit (if they asked for one) if the ship does not port for some reason.
Coasta Maya- If you are still up for touring by this point... walk all the way out of the port store area. You will find tours guides (they can't come into the port area) outside you could book a tour from. There really isn't much there except for Mayan ruin tours (make sure you have enough time) or beach breaks. The beaches along that coast are excellent.
Hope that helps. With a little web searching you can book any of these ports on your own. I suggest looking through the cruise line tours and picking something that interests you and then go on line and search for it. Most anything you can find through the ship you can book on your own in these ports. Hope that helps. Do some investigation, when you have questions come back here and we will help.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013