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.
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