I think you will have many more options if you leave from Florida. Most of the Caribbean cruises leave from Ft. Lauderdale or Miami. Quite a few leave from Puerto Rico as well.
I'll comment on my experiences:
I've been on 4 cruises. Two Celebrity (from Ft. Lauderdale to Eastern and Western Caribbean), one Seaborn (from Rome to Athens) and one Royal Caribbean (From Puerto Rico to the southern Caribbean).
Celebrity and Royal Caribbean would be within your budget for many of their Caribbean cruises. I've had great experiences on all the ones I've been on, but I'd go with Celebrity if I had to choose again. They both have the same types of service and ships. Everything but alcoholic drinks and side trips are included in the fare.
Seaborn is probably going to be too pricey for your budget. Most of those start out at 2-3,000 dollars for a week and leave from Europe. However, they include all drinks and many excursions. The ships are also a lot smaller and the service is out of this world.
Some tips I would suggest regardless of what company/trip you choose:
-Go for as long as you can. Don't even bother for the less than 4-5 day cruises.
-Book your excursions as soon as you get on the ship before everyone else does.
-Opt for the late dinner time if given the choice.
-Pack a change of clothes or two with you in your carry-on in case the airline looses your bags. It's a nightmare to have your bags meet you in the next port. It takes a day or two AT
LEAST for them to catch up with you.
-Rooms with a window aren't worth it unless you plan on spending quite a bit of time in your room during the day. You can't see much of anything at night.
If you want to talk in more detail I'd be happy to.