A few thoughts . . .
I have not made the passage you are considering; but I am soon to make one from Florida (Fort Lauderdale) south to the Panama Canal. Therefore, I have been doing a great deal of reading and research
I am sure you will get a great deal of assistance from others on this web site who have previously sailed the course you are planning.
I can speak to the Florida issue . . . not Texas . . . for boatyard work. (I live and sail in Florida, as a home base. Actually, I have lived on both coasts of Florida for more than 20 years, so I have a fair amount of experience with boatyards and suppliers.)
Fort Lauderdale/Miami is a U.S. yachting center. In Fort Lauderdale, you will find everything you need in terms of sailing hardware, and every service you need. The downside is that Fort Lauderdale also is a "megayacht center," and therefore prices can be higher.
One advantage I found in Fort Lauderdale is that there are at least two boat supply companies that do a substantial business in "used" or "recycled" equipment and parts. I have been surprised at their selection of good quality items at low prices. So, if you like shopping for items for your refit, the availability of these companies in Fort Lauderdale is a bonus for you, and perhaps a money-saving option.
If your goal is to sail across the Gulf of Mexico to Texas, then another option for your refit (if you want to do it before Texas) will be the Naples/Fort Myers Florida location on the west/Gulf coast of Florida. Naples is expensive (but has an excellent city dock). In Fort Myers, you will find labor that is less expensive, if your refit is labor intensive?
As to the northerly winds. Even though you are now in the Caribbean, it is about to become winter up here (Florida) . . . that is as much winter as there is in typically sunny, warm South Florida. As you undertake your weather research, you will find that winter brings cold fronts moving south, and heavier winds. And, yes, some of those weather fronts/winds can be very challenging. The trick is to "wait them out" and pick your weather window(s) for sailing.
It varies from year to year, but the fronts typically begin in mid-December and continue through January, at least. The fronts start diminishing in February, or so, and by March going into April, historically, are not such an issue.
If you are planning an "island hop" around the Antilles northward, you'll probably be past the winter fronts season by the time you get to a northern location and make the hop to the U.S. If you are planning a more direct passage, perhaps Curacao to Jamacia (or Caymans) and then around the Western tip of Cuba toward the U.S., leaving in mid-December may not be a great (weather) choice. I am not as knowledgeable as I would like to be about the effect of the winter fronts on the southern Caribbean sea?
FYI, I've also read (but not experienced, first hand) many different opinions on the choice of a route around the western tip of Cuba, or the route between Cuba and Haiti.
I am hopeful this posting will be a "starter" for more posts from other sailors who can offer first-hand experience to you.
And, by the way, if you need an extra hand on board for part of your passage, let me know. I am a retiree, a 40+ year sailor, and have very flexible time in my life.
And, if you choose Fort Lauderdale as your U.S. refit location, definitely send me an email. Perhaps I can help in some way. It is always pleasant to meet fellow cruisers. You can reach me directly at [jesfl45 at yahoo dot com].