Congratulations on your pregnancy!
I have never been to Jamaica, so I can't speak to that. But I do agree it would be a shame to not get any feel for the local culture. Even just a taxi ride around town might be fun. I've often found the taxi drivers make the best tour guides! They know all the interesting insider things about the area and are happy to talk about their towns. We had a fascinating ride through some of the poorest regions of St. Lucia (similar to the poverty in Jamaica). I felt very safe the whole time.
While I haven't been to Jamaica, I have been pregnant! Honestly at 11 weeks I wasn't in the mood to do a whole lot anyways. The fatigue alone can be exhausting. Expect to need a lot more sleep and rest than you are used to. I like to drink (booze AND coffee) too, but during the 1st tri I couldn't stand the thought of either of those things. Chances are you won't even miss it. Also, that early on it might not be an issue, but I've found that it can be a pain to travel when pregnant because you have to pee SO often! Having to find a public restroom every half hour can be a pain. Better to spend time in one place where you can easily get to the bathroom.
Savor these last days of downtime while you can! Pretty soon you won't be able to just lay by the pool reading a book, blissfully unaware of what else is going on around you : ) And, be sure that you read a good nonkid related book! They'll be plenty of time for parenting an childbirth books later on.
I'd be careful with what you eat/ drink though. As fun as it is to try new things, during pregnancy might not be the best time. If you were to get any kind of food poisoning, or "traveler's stomach" it could be really dangerous.