Hello.....The short answers are YES and YES providing you do your homework (or rely on some else who has). Heathe.
There are quite a few excursions that it is difficult if not impossible to do independently. But generally speaking you can do you own thing in most Caribbean Islands but simply hiring a taxi from the quayside. This is considerably cheaper than the ship excusions, especially if you share a taxi with others. If all you want is a beach to lounge on or a trip to see the sights on a island then go ahead and do it yourself. Just ensure you get back to the ship in plenty of time.