Another vote for St. John. If you have never been to the Carribbean you will very much enjoy the experience. You can take a ferry out of Charlotte Amelia near STT or if you rent a car you can take the car ferry out of Red Hook. You can find a room on STJ in your price range if you do some digging, I have stayed in an efficiency villa for $50 per night up to a 12 bedroom lux villa for much much more so I know it's possible, nothing beats STJ and I have been all over the Carribbean.

SXM is great but it can be a bit expensive, best choice is to rent a car and go to the beaches on the north side of the island that take a 15 minute walk to get to, they are absolutely amazing but I'm not much for over crowded touristy beaches so I steer clear of Orient and Simpson, YMMV.

On your budget you can defi Italy visit the Dominican Republic, Aruba, Jamica, the outer islands of the Bahamas, etc but it will take a little work. Best is to figure out which islands you want to visit and then check out vrbo.com for lodging and the forums on trip advisor for good info.

Good luck, the Carribbean is amazing, diverse and every island is so very different. Sounds like a great trip you have planned.