In Grand Cayman you can have a taxi driver take you to one of the local hotels on 7 Mile Beach, that accept cruise passengers. He'll know where. They will let you use their lounge chairs, etc.
I haven't heard of any beaches in Jamiaca, but there is a beach area by the falls..
While in Grand Cayman, I would highly suggest trying to get to Rum Point. Not really easy to arrange transportation, but it can be done with some investigating/searching. We thought this was one of the most perfect beaches and settings that we have ever been to. It is so beautiful, not crowded, the water is so calm and clear and so many other great things about it, it is a shame to miss the beauty of it.
I have to agree about Rum Point, nice place to spend an afternoon, just totally relaxing and beautiful. There is some decent snorkeling there as well, a restaurant/bar and plenty of lounge chairs too.
In Ocho Rios your best bet for beaches probably would be to get a day pass to one of the many beach front resorts, or when you get to the pier talk to one of the bonded taxi drivers, the ones with the vans with red tags on them ask them to take you to a beautiful, safe beach, and arrange for ROUND TRIP transportation, and pay them when they safely return you to the pier
We went to cemetary beach. It is very small, no tourists around, very good snorkeling not far away from shore and very beautiful. However, there were no lounge chairs for rent.