Be very careful about NYC. You SHOULD be ok if it is a connecting flight, but if for some reason the connecting flight is late or cancelled, taking possession of the shotgun in NYC will likely get you arrested. Both NYC and New Jersey do not care about the Federal law that is supposed to protect gun owners while traveling.

Keep in mind it does not matter what airline policy is regarding firearms, it's what the police decide to do that matters.

I have become addicted to English hammered shotguns to the detriment of my wallet.