My 2pence. I haven't read into many of the replies and so I apologies if this has already been said.
The problem with what you are talking about doing in your OP is that there are not many people wandering this country, hiking and camping and so you stick out like a sore thumb. Everywhere you go you have your belongings on display, advertised to anyone who will see you, which will include those opportunists. Although Doms don't see many people backpacking around they will understand that you are and so have ALL your belongings on you, and so will understand that you will have your bank cards on you which could cause you a brief kidnap to the local ATM in some sleepy village you hope to experience. It happens.
It isn't very easy getting around without transport also, there are plenty of links on public transport but you will be doing a hell of a lot of waiting around, arriving in a town late and then trying to make your way out of town without drawing attention.
You want to spend time up in the mountains, in the smaller villages, you know how bored you are going to be, how much money you will need just so you can sit around outside some colmado and have a drink just to pass the time. There is absolutely nothing to do in these small villages and transport out to any point of interest will cost you.
You might also want to consider where you will be leaving your kit while you head off on a sightseeing trip, I wouldn't trust anyone and I wouldn't have enough confidence no one was watching me stash it.
Camping is romantic and Camping the Caribbean even more so as an idea, but you need to understand that around you are people who are ruthless in what they will do if there is a payday at the end of it, and just as much in rural areas. People in general are used to a more gentle pace of life, but the violent nature here makes it so much more of an undertaking than say Europe, that must be taken onboard because I get the impression you think you can judge this place by previous experiences when infact you will very much be a fish out of water.
At least have back up funds to get you into a hotel once you realize what you are doing and how other worldly the hiking/camping thing is here. There is a very very good reason the sides of the roads on this pretty country are not lined with students taking a time out to explore.
Saying all this there are ranches everywhere that will put you up, for a price, and your budget will not stretch to that, out of every 10 people you ask to do you a good turn here, possibly 1 or 2 will oblige, get your wallet out for the rest.
A wonderful idea, but not such a great reality imo.
Good luck!