I grew up near Ft. Lauderdale and have taken several cruises (before I had children). Honestly, I think they are horrible and have no desire to take one with my children. The last one I took was in 2001--a pre-wedding cruise with mom, sisters, aunt and cousin. We had 3 of the most expensive suites, which also entitled us to dinner with the Captain. I think it was on Norwegian. I would be WAY TOO paranoid to take kids that young. I would be afraid they would get hurt, fall overboard, or even get sea sick. Also, the cattle call of getting on and off the ship would be horrible. I know a lot of the ships do "kid programs", but I personally would never let my 3 or 4 year old out of my sight on a ship. I just couldn't trust a cruise employee to watch out for my child.
With that being said, if you wanted to go, I would suggest spending the money for a Disney cruise. I know they're not cheap, but they are probably the most kid friendly (no grouchy senior citizens would go on a Disney cruise) and have the most to offer kids/families. A friend of mine took one with her girls when they were 6 & 7 and loved it! Also, I know the rooms with balconies are nicer, but I would be afraid about what could happen to the kids (I'm normally not such a paranoid person--just when it comes to water and kids).
The Caribbean is beautiful. We honeymooned on St. Maarten and have traveled to other islands.
Did you ever consider just going to Disney? Many of the hotels offer babysitting so parents can have a night out. Or, maybe you can bring along Grandma, so you can go out at night. Just a thought...