For the shopper, when in Cozumel, head directly for Los Cinco Soles. It's a wonderful, clean, collection of stores and stands offering everything from pottery, blankets, and general souvenirs. There's also a great restaurant out back, Pancho's Backyard, that has fabulous margaritas and guacamole.
Of course, Cozumel is a snorklers' paradise, so the shopper may have to go it alone... but it's perfectly safe within this establishment.
In Grand Cayman, you can take a taxi to one of the public beaches along Seven Mile Beach. I honestly don't remember which one we went to, but hopefully someone will come along with some recommendations.