There is no entry fee for Paradise Beach, as the impression is you'll spend money while you're there

You are under no obligation to purchase shore excursions through the cruise line. So, yes, you can just do shore excursions on your own. You don't mention which ports you want to do the activities you want to purchase at. Be advised, however, avis car rent the ship will not wait for you if you are late getting back to the ship. You must be back onboard 30 minutes prior to sailing. Example, if it says you are in Cozumel avis car rent until 7:00, you must be back on board at 6:30 ship time.
This brings up another issue. Make sure you check the daily planner, avis car rent and listen for announcements whether or not to change your watches. It is up to the Captain whether or not to change the ship time to match the local island time. If they do not, the ship time in some cases will be one hour later than local island time, so be sure you keep your watch on ship time, otherwise, you will miss the ship, and have to catch up at your expense.
In Cozumel, you will probably have to take a cab to one of the beach clubs. They are right at the pier when you get off the ship, and there will be cabs waiting at the beach clubs to take you back to the ship. The one beach club I went to that I liked was Paradise Beach. I think it was about a $15-20 cab ride from the ship. They take US dollars in Cozumel, so don't bother exchanging your money.
There is no entry fee for Paradise Beach, as the impression is you'll spend money while you're there. Lounge chair use is $2 US, and use of their aqua park (large inflatables that they put in the water) is $12 US. There is a restaurant avis car rent on the premises as well for lunch, and for drinks.
In belize there is a market place where the ship comes in and you can walk around the plaza and check out all the various vendors and I believe there is a pool area as well as a place to swim nearby. When we were there we made that stop a quick one and headed back to the ship and enjoyed the luxury of smaller crowds in and around the pool area.
In Belize, you have to tender into the actual city. This means, they use smaller boats to take you from your ship to the shore. Belize would actually be a place I'd do an excursion, either through the ship or through shoretrips. The snorkeling there is fantastic. In Costa Maya, you can grab a cab to go downtown, actually it's to the actual city which is a quaint little fishing village. We went to the Tropicante avis car rent and had a great time. We were able to get a table and chairs, they also had loune chairs available. They had great service, you never went hungry or thirsty, and they ran a tab for us. The restaurant/bar itself is across from the beach, but the chairs are right on the beach. It's a great way to spend the day, good snorkeling, too.
We were just in Belize last week and booked a cave tubing trip through Reggie Tours. We had a blast!! Dennis picked us up at the dock and we were the only ones that had booked with them, so we had a private tour. Dennis was great. He was so much fun and taught us a lot about Belize. After tubing we meet his wife for lunch. We got so much more than others we talked to who booked through the ship.
I do not remember the name of the restaurant. It was a place that Dennis avis car rent recommend. He said that it was one of he and his wife's favorite places. It was only a few minutes from the cruise terminal and had outside water front dinning. We enjoyed it and very reasonable prices.