If you were driving, I'd say put it unloaded in the trunk and drive, but since you are going thru NY by air, a state where FOPA is routinely ignored by some departments and agencies, I would ask you how much risk you are willing to take.
Under FOPA, so long as you are legal at your start point, and legal at your destination, you are supposedly allowed safe passage for a properly declared and prepared firearm ( in the proper TSA container, ammo not in the same container, etc etc etc), and in theory you would be legal to transport unloaded in Ct to your destination where you don't need a permit to have a weapon in your home.
If something goes wrong are you in a position to get lawyered up, post bail, and defend yourself in court?
Personally, and UPS and FedEx can go eff themselves, I'd break the weapon down put the parts that do not constitute a firearm in my checked bags ( slide, barrel, mags) and I'd ship the frame to myself UPS second day air with tracking without telling UPS what was in the box.
There is no need to declare gun parts in your checked bags
Or you could mail the slide and other parts to yourself USPS