You havent looked very hard
There are plenty of place to launch in CT. Old Saybrook has a nice big public launch under I95 - free, Guilford has a public launch ($30 pass for the season), Old Lyme has a nice quiet launch, free, on the CT River. I've launched out of Stonington too.
Lighthouse Point in New Haven is wide open with a big parking lot and free for boaters.
I havent launched south of New Haven, but I bet you could find a spot here
http://www.lisrc.uconn.edu/coastalaccess/ If you are talking about mast up beach storage, then that's a differient story, you either need to join a yc or find a private spot to rent, or make friends with a person who has beach access.
Roton Point YC has a large Cat fleet, mast up storage.
http://www.rotonpoint.org/sailing/index.html The issue with Long Island Sound is the wind, its light during the summer the farther south you go,
Head north on 95 for better wind.
Bill