Based on your time frame selected, not much of a difference as most "kids" are still in school. Kids have their own area (kids' program) anyway so I can't see kids jumping around everywhere.

Just a point that I like to make, we have criused quite a few times and have doen so for more than 20 years. We, DW and me, have never ever used a pool or a hot tub on any of the ship, regardless of cruise lines. We always packed our swim suits, but have never used it on the ship. Not once.

Regardless of when you cruise, ship almost always sail with full capacity. The ships are large but it will be less crowded than you think. The execptions maybe prime feeding time, boarding and disembarking.

If you have never been to Jamaica before, Dunns River Fall is probably a good destination, about an hour away. Beware of toursit traps though as it is getting really bad in Jamaica. They have recentkly tried to clean it up somewhat with "tourist police" (No they are not cops but that's what I called them, they check on vendors and make sure the vendors are licensed, and tourists did not get taken advantage of.) You can always take on the local beaches, city tour (colonial buildings etc)

We have been to Cozumel several times but we took the ferry to Playa de Carman more often than not. If you have never been there, Tulum would be first choice, Xcaret is second choice and Coba would be my third choice. There are several Mayan ruins on the island such as San Gervasio. They are hidden gems and well worth visiting if you are staying on the island. There are plenty of beaches as well. Decent flea market sralls right at the dock.

For me, Nassau is pretty boring but other people just loved it. Over crowded with tourist, expensive shopping. Yes, there are nice beaches but many other places have nice beaches also. Many just spend the day wandering around town, or buy a pass to use the resort (example: Atlantis)