Since Labadee is a privately owned resort you have no choice but to use the tours sold by RCI. No local operators are allowed.
In Jamaica I strongly urge you to also stick with the ship's tours. Too many shady characters offering tours there - it isn't worth the chance you'd be taking.
In Grand Cayman I think Captain Marvin's is a good operator.
On Cozumel you really can't go wrong. It is perhaps the easiest place to get around on in the Caribbean and all the tour operators and cab drivers are safe and trustworthy.
Also - we ask that the same topic not be posted more than once by the same person. Therefore I have removed your other post since it is asking the same thing.
I do wish to note that looking for non-cruise ship tours in order to get around age limits for kids may not work. The ship's tours are run by local tour companies too, and thus the rules are often the same regardless of who is selling the tour. They don't want the liability anymore than the cruise line does.