Hi Clas and welcome to CruiseReviews! As everyone has stated, Jamaica is the only port where I do not recommend doing anything on your own. However, with 13, you have your own group
Seriously, though, if you wanted to do something like an all-inclusive resort day pass, that might be more affordable. I've not ported in Falmouth but have in Ocho Rios, Kingston, and Montego Bay. The majority of the Jamaican citizens are wonderful, kind people. However, in all the ports I've been to, there are the few that ruin it for the many.
Grand Cayman and Cozumel are ports that you really don't have much to worry about doing on your own. StingRay City is the main attraction in Grand Cayman and is really a very unique experience. You need a boat to get there. I'm not sure how much can be saved by booking independently, since I've only done ship excursions there. Shoretrips.com has been highly recommended by many people on this board. My favorite non-expensive thing to do in Grand Cayman is take the bus down to 7 mile beach. It's absolutely gorgeous there. Abanks used to let you use their platform to snorkel from. They're to the left hand side of the tender area (facing the shore).
Cozumel, Paradise is free to get in. I prefer Chankanaab because there is a lot more to do. They have the dolphins, snorkeling, zoo, gardens, shops, water toy rentals, and really clean beaches. Paradise is nice, but the snorkeling isn't good and to me, the beach isn't as pretty.
Hope y'all have a marvelous time!