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In short, several good options for shore diving Grand Cayman close to the cruise ship terminals. Get ashore, turn south (water to your right) and walk. Five shops within one mile: Abanks, Eden Rock, Don Foster's Casurina point, Seaview, and Sunset House.
If you want to take a ride, catch a bus to the Turtle Farm. Next door DiveTech has a shop in the lower level of The Cracked Conch restaurant.
Any of these shops can rent you all of the gear you need to dive.
For Belize, you're probably looking at going with the ship's operator. Boat diving is bigger bucks, so possibly look for other choices.
Numerous operators to choose from in Coz. you don't have to go with the ship's op there.
Roatan you can go either way. Some of the resorts have great shore dives.