The Florida Keys unquestionably have the best sailing conditions in the United States.
However, I'm afraid there is not much of what you are looking for. First of all, there are very few beaches here. The Keys were originally a coral reef, so they are all made of rock. And most of the Keys are protected on the ocean side by reefs and on the bay side by mangrove islands, so there is not much wave action to create sand or wash it in from the ocean.
There are two places that have both beaches and camping -- Long Key State Park and Bahia Honda State Park. However, you are not allowed to keep your boat on the beach, so you would have to pull it out via the ramp every night. I don't know whether you would also have to drop the mast every night.
Most of the other beaches in the Keys are manmade. Most on-water motels and campgrounds advertise that they have beaches, but they are usually more like "sandboxes" -- they are separated from the water by retaining walls.
America Outdoors, on the Florida Bay side in Key Largo, might be your best bet. It is popular with the Jacksonville fleet, which usually makes an annual pilgrimage to the Keys. The campground has a big beach, but, again, it has a retaining wall between sand and water, so it is basically useless for the purpose of beach cats. But what those sailors do is anchor their boats in the water just out from the retaining wall. It sure beats pulling the boat out every night and de-rigging it, which is the only other option there.
The other thing to consider in the Keys is that the best sailing and island-hopping are on the Bay side, but the best snorkeling (reefs) is on the ocean side. And there are few places to get back and forth between the two (Channel 5 Bridge, Marathon, and Seven-Mile Bridge. So, for snorkeling, you probably want to find a place to stay on the ocean side, and for island-hopping, you probably want to be on the bay side.
If you plan to use your boat to do the snorkeling, you could camp at John Pennekamp State Park in Key Largo. It would involve launching your boat on a ramp, sailing/paddling out from Largo Sound through a winding mangrove channel (along with all the dive boats, snorkel boats, and fishing boats) and then sailing about three miles offshore to get to the reefs.
Another possible campground on the ocean side in Key Largo is Key Largo Campground, 305-451-1431. Again, you would have to use a ramp. And I am not sure whether you can get to the ocean from there or not with a mast -- again, you would be wending your way through mangroves, and I don't know whether any small bridges are in the way.
Another option on the Bay side in Key Largo is Kings Kampground, 305-451-0010. They have a few tent sites. Again, you would have to use a ramp, but maybe you would be able to at least keep your mast up overnight.
Also, I don't know what you consider camping on a "shoestring," but campsites in the Keys average about $35 a night. (I don't know what the cost is at the state parks.)
If you really want shoestring and adventure, launch your boat in Key Largo at the Caribbean Club and load it up with stuff, and sail it to Nest Key out in the backcountry, where you can camp for free (just get a permit first from the Everglades Park Rangers -- it used to be free, and I don't know whether there is a nominal fee now).
If you are going all the way to Key West, try Boyd's Campground, which is right before Key West.
As I said, the Keys have the most beautiful sailing waters you will find anywhere. The problem is the point where water meets land.
As somebody else said, please let us know what you find out. We live here, but sometimes tourists find new places we don't know about.
If you have more questions, please feel free to e-mail me,
mary@catsailor.com 
