For the ports you are visiting, you can pretty much "do your own thing" without too much reservation. Nancy is right, though, realize that if you miss the ship, you are on your own.
I would plan a cushion of at least an hour to make the last tender or departure time. This way, you are almost assured of making the sailing time.
Virtually all of the taxi drivers will speak some English (for sure in Grand Cayman) but learning a couple of words in Spanish, will help dramatically.
Have a great cruise!