I'm not far from Hartford, so when you make the move be sure to look me up!
Much of the private land is off limits to hunting, either b/c the owners don't like hunting or the land is leased by a hunting club. There are exceptions, you just need to do the leg work to find them. I wouldn't expect to pay to hunt, just maybe help out and supply a few packages of venison.
There's plenty of public land, some of it real nice for deer and tree rats. Many smaller tracts of land, some larger. You'll share much of it with hikers and birdwatchers, but the thick stuff (ie where the deer are) you'll have to yourself.
Overall the deer population is out of control. Most of the state is way beyond the carrying capacity of the land. Seeing deer has never been a problem (even the big ones) but seeing them on land you're hunting and not just over the property line can be frustrating :D