We just sailed the western Caribbean itinerary in November and did two private excursions that I would highly recommend. In Cozumel we did a snorkeling tour was operated by the Playa Mia beach park (we booked it through Costco, but you can book directly with Playa Mia as well). It departed right near where the ship docks where you started with a nice boat ride along the coast and made made two snorkeling stops; first at a coral reef where we saw tons of beautiful fish, and a second stop at a shallow sandy area filled with starfish. At the second stop it was shallow enough to stand, so after a little while of snorkeling they started serving drinks in the water and we all got to hang out on the sand bar and have a few margaritas before boarding the boat again and heading to the Playa Mia beach park for a late lunch and some additional fun in the sun. From there you could depart whenever you were ready and they would pay your cab fair back to the pier (a short ~10min drive). I did it with my 11 yo son and we both had a blast, and would recommend it to anyone (and will probably do it again with the whole family if we sail the western itinerary again). I would also note that there were lots of other people from our ship on the excursion, so it might be the same place the Disney contracts with (but a lot cheaper if booked directly).
In Jamaica we also did a private tour (again booked through Costco) to Dunn River falls (I apologize as I don't remember the individual tour company's name). It was a great to do this excursion with a small group instead of a large bus tour through the cruise, as we arrived at the falls on the early side (well before the main rush) and got back to the ship early afternoon. The falls themselves were a lot of fun, and a very unique experience that I would recommend to anyone (at any age) who hasn't done it before.
Now that I think about it, Costco Travel actually outsources their excursion offereings, so I think anyone can book them through a company called Shore Excursions Group. I'll definitely be using them again when looking for Eastern Caribbean excursions on our next cruise. Best of luck, and enjoy!