I'm sorry to say, there isn't much in the way of camping here period, pretty sure there isn't anything open year round anyways. Most places close up shop at the end of October, that includes the Charlestown Breachway (which is dry camping BTW)
There is
KOA Mystic that is actually in Stonington, just outside RI on I95 that's open year round. Not sure where in Rhody you're headed, but this would give you easy access to most spots here. It's only one exit from the RI border.
Good Luck!