Have you looked at the USVIs? Here's a link to the webcam at beautiful Sapphire Beach on St Thomas just to give you a taste:
Sapphire Beach, St Thomas
and here's the webcam link looking at the ferry dock on St John with St Thomas in the background:
http://www.stjohnspice.com/spicecam/
We don't have a "rainy season" as such and historically the greater amount of rain falls in May and November but it's extremely rare to have days of rain on end. It's a good idea to take out trip insurance wherever you travel in the Caribbean during official hurricane season which is June through November. It's very inexpensive.
There are all sorts of accommodation options to suit your budget although very little all-inclusive as these are islands where getting out and about exploring is the thing to do!
If you'd like some links to websites and forums which can provide you with information I'll be happy to help out.