Fishermen's memorial state campground in Narragansett. Tough to get a site this late in the season but maybe you can get in on a cancelation.
Beaches down the road, centrally located about 40 minutes drive give or take from Mystic Seaport, Ct & Newport RI.
If you bring road bikes you can ride them down to the ferry (5minutes away) and spend a day on Block island which has touristy shopping, restaurants, and of course biking. I believe you can also rent mopeds or bikes on block island. But its worth a day trip.
Hope this helps
Mike