Dear Mimi:
I'm the family cruise editor here. You should read my article "Back to the Beach" on this section of the site -- it lists family friendly beaches in the Caribbean that you can get to on your own.
I'm a big fan of catching a cab to a beach on your own rather than spending money on shore excursions in every port. However, Jamaica is one of the few spots where I strongly recommend going on a shore excursion sold through the cruise line. You may get "hassled" by the Jamaicans to buy things from them, etc. if you're on your own. Your sons are a great age to buy the Dunns River Falls shore excursion in Jamaica -- they'll love climbing up the falls. My daughter did when she was 5 and 7 years old.
In Aruba, there's an Atlantis semi submarine which is pretty cool. In Curacao, we did the Dolphin Encounter which was wonderful. Your sons are a good age for that too. Aruba's Eagle beach is nice to go for a swim. Curacao's main town of Willemsted is good for shopping.
I hope this is helpful. Let me know how you liked your cruise after your return.
Smooth sailing,
Luisa Frey Gaynor