I up and moved to St. Thomas, USVI because I wanted to stay closer to the East Coast. I moved for the exact reason you mention - why not? Moved down without a job and two days later had an interview and two days after that started working.
I worked afterschool at a private school. Job was 3pm-6pm on weekdays. I could walk there, and the pay covered my (expensive) rent and basic groceries/laundry/etc. I was not living high on the hog, but was paying the bills.
All in all I stayed on island for a year. It was really nice, but being far from everyone gets to you (it is a long flight just to Miami, much less any further inland) and you can only go sit on the beach with a few hundred tourists so often. The resident population is very divided - very low income and very high income, with little middle class. A lot that fall into the low-to-middle class tend to be highly transitory. After I had been there for over six months I almost became an "old timer" on the island.
I am glad I did it, but unless I came into a good amount of money I am not sure I would move back.
I am not sure what your nationality is, but a lot of Caribbean islands "belong" to various European countries. May not be difficult for you to move there.