Not intending to be disparaging towards Caribbean dive destinations, since I dove there for many years and still dive there on occasions, but I think that after one trip there, you may decide that you've seen all that you wanted to see, especially when you compare them to the dive destinations that you have done in the past.
The Caribbean, for the most part, has very clear and beautiful waters along with lush coral reefs in some places, but because it is a younger body of water than the Pacific, it will never match the the colors, the diversity, and the abundance that you have seen in the Red Sea and the Indo-Pacific.
As far as liveaboards in the Caribbean and general vicinity, I would say the Caribbean Explorer that visits Saba, St. Kitts, Nevis and maybe Statia will probably give you a very good sampling of Caribbean diving, with a good mix of pinnacle, wall, and coral garden diving. Turks and Caicos Aggressor and Cayman Aggressor are also good, and if you want to venture into Central America, there are the Utila Aggressor and the Belize Aggressor. There are liveaboards that dive the Bahamas, altho I'm not too crazy about them because they pack waaay too many people in them.
This area does offer a number of unique underwater experiences that are quite appealing to divers from all over:
(1) snorkeling with humpback whales in the Silver Banks between the Dominican Republic and Turks and Caicos between January and March. You can do this on the Turks and Caicos Aggressor, one of the Nekton boats and the Peter Hughes Wind Dancer.
(2) snorkeling with wild spotted dolphins in the Bahamas. There are a number of boats departing out of Florida that do this, along with one boat out of Bimini. One of the sister boats to the Caribbean Explorer offers a trip that combines some diving with some dolphin snorkeling.
(3) the shark feed dives in the Bahamas, where it is possible to see tigers, great hammerheads, lemons, bulls, and on some trips, oceanic white tips. There are three liveaboards that I know of that depart from Florida to do these types of dives.