Is there anyway you could increase the time and budget to allow for a week-long trip? Realistically, with 5 days you will probably only get 2.5 days of diving because you will be traveling 2 days and you also have to wait 18-24 hours after diving before flying.
But if you could spend a week that would open up more destinations, you referenced Anthony's Key but it isn't always easy or cheap to get to Roatan.
If you are definitely limited to 5 days then you should consider destinations that you can reach quickly and easily (nonstop flights would be best) because you don't want to spend a lot of time traveling, and if you run into any connection delays they will eat up even more time. You might want to add Florida to your list of possibilities.
You mentioned "April or May" but I think that it would be best to travel after Easter/Spring Break is over because they are in the high season. The prices usually start to drop in May (or at least late April). The water will be warmer also, and that is desirable IMO for more northern destinations like Bahamas and Florida.
I know that you can fly direct to Cozumel from Houston. It is an economical option with excellent diving and the obvious choice when looking for a short dive trip from Houston. But I may be in the minority when I say that I don't think that Cozumel is a great destination for new divers. When I was a newer diver I initially had a hard time in Cozumel adjusting to drift diving and the currents and maintaining buoyancy control. Some people take to it easily - my husband didn't have any problems - but Cozumel doesn't seem like a great first dive trip to me, just my opinion.
My suggestion is at the link below for the Cobalt Coast resort on Grand Cayman because it is good and easy Caribbean diving. You can probably fly to GC nonstop, and the package includes accommodations, food, transfers, taxes and hotel gratuities, and both boat and shore diving - which increases the number of dives you can get for the price. But it is a 1 week package that would cost $3000 for 2 people plus airfare (and Nitrox will be extra if you want to use it) so it may not work for you:
https://www.cobaltcoast.com/grand-cayman-dive-packages/cayman-escape/
Good luck with your dive training and trip planning!