Son, you are unlikely to get to experience 'sailing' in a typical tourist cat in any conditions, let alone during a storm - they truly are designed and run as 'booze cruises'. That being said, there are many inexpensive all-inclusive vacations where you will have free access to Sunfish, Hobies (albeit sometimes only with a staff member
on board). And you can always arrange for a day charter on a smaller boat where your object is actually to go sailing.
The range of hotels and prices is unbelievable - at your total I would look for a moderately priced resort and then leave enough money
for chartering a small crewed yacht for a couple of half-days or so. You can find many private yachts in the Caribbean that willl do day charters at terrific rates; check internet
sites with general information on the islands that you are considering. In addition, you can often approach charter yachts and if they are not busy (likely during your proposed time of the year) get a terrific deal by negotiating on the spot.
The Virgin Islands
are, of course, filled with yachts but short on all-inclusive resorts. St. Lucia now offers a number of all-inclusives as well as available yachts, but probably at a higher price than you are looking for. The Grenadines have the yachts, but no real all-inclusives (and the need for a shuttle flight). And while St. Martin has a ton of yachts, it is short on low cost all-inclusives (and boats will also tend to be more expensive for a private charter).
At your proposed price range I would look to the DR or Jamaica
. I have only been to the DR once (and will likely never return), but there are some very inexpensive all-inclusives and in certain areas, a fair number of sailing yachts. In Jamaica I would recommend a resort in the Montego Bay area as there is a marina with available yachts. As to a hotel
..... check brochures. The Decameron Montego Bay and even the Holiday Inn Sunspree are very cheap
(likely under a thou a head). They are both very close to the airport
(and in the case of the Decameron, right on the 'hip strip' and near some cool bars). All of the rooms in the former are ocean view and it is an inexpensive upgrade to a ground floor ocean front room at the latter. Food
are 'OK' and it will definitely leave money
for chartering a private yacht. Think in terms of $250.00 or less for a half-day for a party of four (you can likely cut costs by finding another interested couple at the resort). Anyway, what you propose is definitely 'doable' - in the past, while I was hunting for properties in the Caribbean I often stayed at all-inclusives as the 'no-hassle' format allowed me to spend the days viewing properties without worrying about the logistics of anything else. There are way better resorts than the ones I have mentioned, but at the price they will provide great bang for the buck and will allow you to budget for some time on a private yacht.