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Matt, the American Red Cross of Greater NY (ARC/GNY) used to have, and probably still has, a dinghy sailing class that is taught on Flushing Meadow Park lake every year. You might want to dig that up, IIRC it is cheap, affiliated with some small sailing club that also has two larger boats operating out of the World's Fair Marina nearby, all accessible from the #7 train.
Most of the sailing schools in the city will be using something bigger, J/24s, Solings, something in the 22-26' length. Some of the yacht clubs have dinghy programs but they're often for the juniors. Still, if you call around you may get some referrals.