Hi, Margot and welcome to CF!
I'll try and take on your questions:
1. Yes, it is possible to hire a
captain (should be licensed, and experienced, of course) for such a trip. There are actually a number of them. Sounds like you are looking for someone who will take responsibility for the trip and also act as an instructor, too. Yes, those people exist.
2. Doing a
circumnavigation in a
power boat is very difficult. Not impossible, but the ones that can do it are very expensive and complex. Think more like a mini-mega-yacht. This is because the long passages require very large
fuel tanks. To accommodate those, the boat ends up quite large, which means bigger engines, simply lots more structure, and so on -- all adding up to big $$$. That is not the case with sail, since your primary means of
propulsion doesn't rely on carrying lots of
fuel and everything to support it.
3. You don't say if this would be on your boat or someone elses. If on yours, then hiring the instructor/skipper would be the way to go. If you're looking to
charter a boat, that could be a bit more tricky, since it would be 1-way, for quite a long time and quite a distance.
Although you didn't ask (and probably already know), I'll say it anyway -- there are a number of pretty darn effective ways to deal with the flying anxiety, and not necessarily with medication, either. Almost any licensed psychologist or clinical social worker can either help you with it, or point you to someone who can. It really is one of the easiest problems to treat, and would certainly open up lots of other possibilities for you.
I also suggest that before you commit yourself to such a trip, that you and your
family do at least a few days of
offshore sailing, first. Preferably to include at least a day or so of less-than-ideal
weather.
Florida to
Costa Rica is a pretty long trip and no matter which way you go around
Cuba, you're very likely to encounter at least some bumpy
weather. It would be better to have an idea about what you are going to be dealing with before you're committed to it.
Good luck with you. Sailing is a great way to travel, but it isn't for everyone.
ID