Cozumel, great for gifts and don't be afraid to have a go at haggling, they love it. You do get a bit of hassle from the locals to get you in their shops but don't push too hard if you make it clear you are not interested. We have been to Cozumel before and must admit we have seen some significant improvements. Again it has a lot to offer and you're spoilt for choice on excursions.
In Falmouth Jamaica the ship docks right into port which is modern and very attractive with the usual shops of duty free, gifts, hair braiding, bars and restaurants. However once out of the immediate port area things pretty much go downhill. The area is very run down and untidy, however the local people are very pleasant and friendly. We had intended to do a planned walk around the area to see what was described as some excellent Georgian architecture which I'm sure would have been interesting but we didn't feel so comfortable so opted to stay in the immediate area, than an early trip back to the ship. A number of people we met had opted for excursions and all said that although they enjoyed the excursion (which there are many to choose from) they thought the island was dirty with rubbish lying all over the place. Perhaps the cruise lines going there might encourage the parish to put some remediation plans into place.
Grand Cayman is a place we have visited before and definitely worth a visit. Probably the place also to buy your duty free alcohol. Smirnoff $10, Brandy $16 for 2 bottles, Gordons Gin $12, Jamesons $12. Plenty of excursions and you can always go to 'Hell' and back. Try and go somewhere near the waters edge and you are bound to see huge Tarpon swimming really close to snorkelers and divers.
Grand Stirrup was a lovely day on the beach. It seems to be still work in progress but you can imagine how lovely it will be once the plants have matured. Great for snorkelling and diving plus very safe for children. Lots of interesting and varied fish and the colours were amazing. The buffet are was a bit of a bun fight with long queues, we didn't bother to eat so can't comment on the food. The island is a bit shy of shaded areas but I guess this will improve. All in all it was an enjoyable day and would go again.