Costa Maya seemed to be a wholly owned subsidiary of the cruise lines. I can't speak to the shore excursions as we did a little shopping in port and then gave the rest a miss. Some of our cruise companions did go into the actual town (which is a bit of a ways away) and hit one of the local restaurants for a lazy lunch and some beers.
In Belize we gave the shore excursions a miss and instead we booked a snorkel trip right in the port for $49 per person. They take you on a boat (with no shade, pack sunscreen) to a lovely little white sand island. You can hang out on the beach and drink rum drinks at no extra cost before and after you snorkel around the island. Everyone there was a cruiser and they were very mindful of when the ships were leaving, so there were no worries.
Cozumel has grown a lot in the nearly 20 years since I started going there. I recommend going to Chakunaab park, which is where you can snorkel, see the dolphin, get in the water with the dolphins, etc. You can either book through the ship or take a taxi there and pay your own way in (not cheap, basic entry is a bit over $20 per person). The park is clean, the employees are polite and even if you decide not to pet the dolphins, there is a free tour of the dolphins given by the staff that is actually really nice.