The only "public" range that I can think of in that part of Connecticut is the CT DEEP range in Glastonbury, which has limited hours hours and works on a reservation basis when open.
There are a number of nice clubs to look at: Rockville and Andover are two of them. Rockville does do a lot of public activities (rifle, pistol, and shotgun) that are well run and affordable.
If you are willing to deal with Massachusetts legal nuances, Springfield Sportsman's in Monson (just over the line from Stafford Springs) is a great club at a wonderful rate- with no work hour requirements. I'm a member there and recommend it.