The USVI's/BVI's tend to be shallower, easier diving. A lot of Cozumel is deeper drift diving, maybe not the best choice for your
first ocean dive trip. Of course you'll need a passport to go over the to BVI's also. One option might be to commit to the USVI's and send for your passports via the standard process - if they arrive in time - you can go over to the BVI's on the fast ferries also - it would be fairly easy to coordinate a day trip over there - some days it would even be possible to schedule afternoon boat dives and still get back. Or spend a few days on Tortola, the ferry ride is 50mins. It's easily possible (and cheaper) to book an afternoon return flight from St. Thomas also - some friends were able to get back from Virgin Gorda with time left over to look around/have lunch downtown before their 2PM flight.
Some of the better diving we did was in the BVI's a few years ago - not as many dive operators so there's less pressure on the reefs. Most times we saw sharks, turtles etc. they were pretty indifferent. And it's really stunning topside, beautiful beaches, lots of small islands to explore, there's a lot of day sail options there - most things move between the islands by boat - the ferry schedule will give you some idea of what's possible.
Virgin Islands Ferry Schedule
Or look at St. Croix. It's the U.S., there's excellent shore diving both off the Frederickstad Pier and Cane Bay wall on the north side. There's also boat trips out to Buck Island National Marine Park from there. From DFW I'm fairly sure there's still a 1-stop AA flight thru MIA.
St. Thomas has a couple of shore dives but mostly it's more expensive boat diving there. Either place expect to pay more for accommodations since it's high season now. If you do go to St. Thomas, you might stay in the East End - you can dive both Sapphire Beach and Coki Beach from shore. Sapphire Beach is really shallow though, Coki Beach is where the cruise ship divers go. It's decent reef diving not much over about 60-80' IIRC.
Both are easy shallow dives and have on-site operators. There's quite a few condo options nearby, if you want to save and cook a little. It is a lot quieter than downtown though. To do some easy wreck diving, look at Admiralty or Blue Island from Charlotte Amalie. We used Dive In at Sapphire Beach once, they were reasonable, have a smaller boat, four of us were their business one morning. I don't think they could take more than eight. We did a shallower drift dive and a second dive off the cays nearby. One of the best dives, Carval Rock, is off St. John but accessible from the East End shops (Red Hook Diving, there's others). It's only divable when conditions are good.
There's also a car ferry between St. Thomas and St. John - runs on the 1/2 hr. So you could day trip and shore dive there also - there's a few more shore dive options there. And a lot of good snorkeling.
On the flipside, Coz can be fairly cheap accommodations. But factor in the cost of all the boat dives and it may not be a deal after you pay to expedite your passports. Villa Blanca resort might be a less expensive option, there's 3 dive operators nearby and they do meal plans. About 2 miles to town from there afaik. There's a shallow shore dive off their property but it's not the best on Coz.
. Also they are two great smaller islands off of Puerto Rico
