So, you're legally licensed in your state of residence, you're moving to CT and you have a firearm you'd like to bring with you.
If that's correct, then I offer the following:
1) Your premier source for firearms info is the SLFU, (state licensing and firearms unit);
Link:
Firearms
2) CT has a couple different types of licenses; essentially ammo permit, longarms permit and pistol permit.
Ammo permit lets you buy ammo.
Longarms permit lets you buy ammo and buy longarms.
Pistol permit lets you buy ammo, longarms, pistols and lets you carry permit, (open or concealed).
3) You don't need a license to posses the firearm; with the caveat that you're on your own property, and there's a carve out for hunting with longarms too.
4) Non-resident permits are issued by SLFU, they have same ground rules as resident permits.
5) Certain firearms and types of firearms are illegal to posses in CT as well as "high capacity" magazines. <spit>.
There are exceptions to these laws for those grandfathered in and for the special people.
6) It is illegal in CT for "prohibited" people to have firearms. See link previously posted.
So - if you're not a prohibited person, and the firearm you want to bring to CT isn't prohibited by CT law, you can bring it to CT. If you do not have a weapons permit and it's a handgun, you can't carry it. I don't know what what happen if you tried to sell it, because sales have to be authorized by the SLFU. If you do sell it, your best bet is to go through an FFL, but you'll lose money compared to a private sale.
Last, but not least - you can transport your firearm from your state of residence to CT under FOPA; and a key component of FOPA is that you must be legal in both origin and destination states. So, make sure you're legal in CT before relying upon FOPA.
I *think* you're legal; but I'm going on what you said - and - I'm a code monkey, not an attorney. You get your ass in a crack, I'm sitting on the sidelines watching - not the one getting hurt. Always verify what you read on the internet when it's your ass on the line.