From your location I'd start with Hawaii. See if you can get a direct flight to Kona. There's the Manta Night Dive, spinner dolphins, the odd shark, turtles everywhere etc. King Kamehameha resort in Kona is part of one of those "suites" hotel chains. There's at least 1/2 dozen dive operators in Kona and most of the boats leave from the harbor nearby. Even a couple of shore dives. Jack's Diving Locker does one at Kona/Kailua Pier regularly and there's others. We like Kona Diving Co. for boat diving.
Or Maui - there's all sorts of places to see sharks on Maui. And turtles. Even a Manta cleaning station off West Maui some of the operators dive. If you get a place up around Lahaina dive with Lahaina Divers or Extended Horizons. In south Maui, we like B&B Scuba, Mike Severns or Ed Robinsons. There's also escorted shore diving with Shaka Divers or Maui Dreams Dive Co. in that area.
Belize might be another option. You should probably see some sharks off Ambergris Caye. I probably wouldn't spend a whole day diving the Blue Hole though due to your limited time. IDK much about Ambergris since we dove in Placencia but the extra flight down and back might take up too much of your time. Belize was a red-eye for us just to get there the following afternoon.
Grand Cayman if you can get there reasonably - time/price wise. You'll see a few sharks, a lot of stingrays, often some Eagle Rays, lots of Turtles - more off the NW side typically. At one time, Sunset House was offering single rates packages that were discounted from their normal prices for 3-4-5-6 or 7 days You can stay there, dive there, even eat there. My Bar is the divers bar and there's a restaurant. It's also a short walk to downtown Georgetown for other options. It's not what you mentioned but the silversides are probably in Devils Grotto right now. That changes that from an avg. - sort of worn dive - to outstanding.
http://vimeo.com/69518760
Maybe the Bahamas? Lots of sharks on Nassau/New Providence. Stuart Coves bundles packages with several resorts and does a shark dive. There's a lot of them waiting for it nearby most days also. Bahama Divers is another option in town - Cove's picks up since they shuttle you to their south side location. You can find some decent lodging in Cable Beach, better lodging on Paradise Island if budget isn't an issue.
Or Grand Bahama. If you can get to FLL by about 2PM, you can often be on-island the same day via BahamasAir. UNEXSO does shark, dolphin and reef dives. It's in Port Lucaya - I think they have a hotel but there's others nearby. Cristina does dives like this there:
http://www.youtube.com/watch?feature=player_embedded&v=WK2LpUoqX6A A reasonable option is Bell Channel Inn, they have their own dive operation.
From Phoenix we can get to Turks & Caicos by about 2PM also. (early or red-eye departure) That's good Pelagic deep wall diving. Better than Cayman or the Bahamas IMO. Stay on Provo and find an operator (Dive Provo, Caicos Adventures) whose going out to West Caicos or French Cay - there's sharks, turtles, rays, grouper there on every dive. Occasionally wild dolphins also. This is a liveaboard trip report but you'll see the same stuff off the day boats - we frequently moored near them.
http://www.awoosh.com/Saudio/Saudio_Turks&Caicos.html <- thanks to Mike Southard...
I doubt you'll find any short term liveaboards unless it's something like Spree or Fling - they do long weekend trips. Or maybe the
California boats. The Caribbean liveaboards are all Sat-Sat - due to the logistics of getting there it doesn't make sense to do anything shorter. Since you're a SF, most of them don't have cabins just bunks.
Of all the places I listed, the most safe is Grand Cayman. I didn't see much evidence of crime on Turks/Caicos either - the captain told us when they throw equipment in the trash it's there a week later unless they mark it "free". Slightly less safe might be Belize City (not Ambergris) or downtown Nassau.
