For a great combo of relaxation and adventure in that budget range, I'd recommend Central America, especially Belize, Honduras, Panama, and Nicaragua. September is hurricane season, but CA's Caribbean coast does not get hit nearly as often as the Eastern Caribbean. You'll likely have some rain here and there, but that's of course pretty typical in tropical areas any time of year!Here are my recommended areas in each country (I would split the trip between 2-4 different areas, depending on how long you spend there).
Belize: Ambergris Caye, Caye Caulker, Placencia, San Ignacio (if you wanted to add a few days in the mountains for some off-beach adventure)
Honduras: Roatan, Le Ceiba, Copan
Panama: Bocas del Toro, San Blas Islands, Pearl Islands, Panama City, David
Nicaragua: Corn Islands, Ometepe, Grenada, San Jose del Sur