While Grand Cayman and Cozumel are great dive locations, I think Nassau offers the best combination of "adventure" diving with lots to do topside (casinos) and is easy to get to from the East Coast.
You can stay at any range of beach hotels west of Nassau (casino) on the beach. There are a range of hotels out on Paradise Island from the Atlantis (casino, waterpark) to the Holiday Day Inn.
My favorite is the British Colonial Hilton downtown next to the straw market. It has its on beach and is next to the water taxi to Paradise Island.
The best part is that all of the major dive operators will pick you up anywhere on the Island (including Paradise Island) so you can skip the rental car.
My personal experience with dive operators is with Stuart Cove's. They are a large and professionally run operation with several boats. They are located on the SW part of the Island away from downtown Nassau. The onsite hotel, the South Ocean Resort is fairly nice and is on its own beach, but it is a bus ride away from the nightlife and shopping.
While Stuart Cove's offers regular one and two tank reef dives, their specialty are the shark dives and DPV wall flying dives. Many of the sites where they visit are along the Tongue of the Ocean, a 6000 ft. trench where the visibility is tremendous and sharks can be encountered frequently.
This is also the James Bond tour also as several of the wrecks filmed in Bond movies (including Thunder Ball) are regular dive sites. Pretty cool.