I am not a water or beach person either. We have only done Western Caribbean and I have always found something to do. I have enjoyed the ruins, tours, walking round, and even playing mini golf in Cozumel. Try the section on ports and see what others have recommended. I know in Coz they even offer cooking classes and some places you can go to an all inclusive club for a price. There you could swim in the pool, have lunch, and just relax. We have only done RCCL and the Rhapsody is small compared to the others, but we always found something to do. Go to www.cruiseclues.com and look at the various ships that you are interested in and what they have to offer. Have fun.
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Rhapsody of the Seas 12/2002
Rhapsody of the Seas 03/2004
Rhapsody of the Seas 12/2005
Explorer of the Seas 05/2010
Voyager of the Seas 01/2011
Mariner of the Seas 04/2012 B2B
Enchantment of the Seas 04/2013
Navigator of the Seas 05/2014