George Town, Grand Cayman, is a tender port, so you cannot really be sure when you will get ashore if you are onyour own because passengers on shore excursions booked through the cruise line generally get priority for tenders. Also, if the demand for tenders exceeds capacity, the shore excursions offered through the cruise line will adjust their starting times whereas excursions that you book on your own will not.
If you go ashore on your own, there are two viable options. First, you can take a taxi to Seven Mile Beach (which, in spite of its name, really is only 5 1/2 miles long). It's one of the world's premier white sand beaches, and it's lined with resort hotels (most of which are pretty expensive). Nonetheless, there are some clubs where you can rent lockers and have access to changing rooms and perhaps shower facilities. Also, you can go to any of the the beachfront bars for beverages or even for lunch. Taxis back to the tender landing are readily available at the clubs and hotels also, so getting back will not be a problem. The other option is shopping in downtown George Town, where there's a large duty-free shopping district right across the street from the tender landing. Prices are not cheap (be sure to add 25% to all prices to convert from Cayman Island dollars to U. S. dollars before agreeing to any deals) but the merchandise is generally of very high quality -- the best that I have seen in the Caribbean. Also, be sure to stop by the rum cake stand (http://www.rumcakes.com
) for some samples (the coconut rum cake is my favorite). If you get ashore at a reasonable hour, you probably will have enough time to go to the beach for a couple hours and to do some shopping on the way back to the ship.
In Cozumel, the ship probably will moor at a pier so you'll be able to come and go at will once the ship gets clearance from Mexican authoriities. If you must go on your own, scratch anything on the mainland from consideration. The Mexican Folkloric Show right in San Miguel de Cozumel is one of the best that I have seen, but I don't know if you can get into it on your own (most ships offer a short excursion that encompasses just this show). The other popular attraction in town seems to be Carlos 'n' Charlie's, a local "watering hole" for those who prefer to stagger back to the ship. There are some beaches and snorkeling areas on the island, but I really don't know much about them.
Have a great cruise!