In Cozumel, I went to Paradise Beach the last time I was there. There is no entry fee, as the impressions are you'll spend money while you're there. There is a $2.00 fee for lounge chairs, and for $12.00, you have unlimited use of their aqua park, floating beach mats, and snorkel equipment. There is also a restaurant and bar on the premises.
I've heard good things about 7 mile beach in Grand Cayman, but haven't been there yet. Maybe one of my future cruises if I go to Grand Cayman I'll check that out. There is shopping in Grand Cayman right at the tender area, but that area I have heard is the most expensive. For bargains, you're better off walking further into town.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)