Excursions depend on your budget but if you can afford a laugh try the river tubing in Jamaica - safe - fun - and you get photos to capture the stupid look on your face as you negotiate rapids on your tyre inner tube in all of two feet of water!
The weather should be great. If you do get any showers they will be over quickly. You need to say which islands you are visiting in order for anyone to advise about trips. The Sea Princess is a lovely ship I'm sure you will enjoy her.