We did the Carnival Legend Western Caribbean out of Tampa a few years ago and loved it! I think it was a little less than 3K for a balcony stateroom, but I'm sure prices have changed since then, and we did a lot of excursions that brought the cost up above that. We swam with dolphins in Cozumel, sting rays in Grand Cayman, snorkeled at a small island in Belize and zip-lined in Roatan. dd's were 11 and 5 at the time and both loved it.
I would recommend visiting Cruise Critic. It's like the DIS for cruising.

They have forums for the different cruise lines, different ports, and also a forum for cruise newbies.
How awesome that you won a cruise!! Have a great time!
ETA: on Cruise Critic, you can find out some of the differences that are important to you in cruising. For example, we aren't into getting dressed up while on vacation. We loved that Carnival allows men to wear dress shorts in the main dining room all but 2 evenings of the cruise. Other people think that's appalling and hate Carnival for it. (and they will be just as vocal about it as some on the DIS are about refillable mugs

) I believe some cruise lines are giving the option of flexible dining times, rather than having a set time for your whole cruise. Again, some will like that, some won't. Read and figure out what sounds best to you. Also, when looking at specific itineraries, make note of the times the ship is in port. Sometimes a "day" in a particular port is only from 7:00am to noon. And when they say 12:00, that means they are pulling out at 10 till, and will tell you to be on the ship by 11:00.