If I am shipping an item from NYC to Connecticut, should I be entitled to pay the NY tax?

If you sell a product within a state, the seller is responsible for the sales tax. If you sell a product to another state and you don't have a retail presence in that said state, the buyer is responsible for the sales tax of the buyers state. If you have a retail presence in the buyers state, then you are responsible for remitting the sales tax in the buyers state.