Two suggestions; firstly check out your roll call of fellow travelers on cruise critic to make 'friends' before you travel- you may be able to share excursions with them - and also check out the ports of call section on same site to see various options.
The Caribbean is ideal to do your own thing. Very often a taxi driver will take you around the island, staying with you all the time. However, you must agree a price with him before you get in the taxi and make sure the price is not per person. We've done this on many occasion and never had a problem. Just make sure you give yourselves plenty of time to get back to the ship.