Verizon will exchange your Target bought Note 7. I have confirmed this through Samsung, and through Verizon.
Whether or not your individual Verizon store will be cooperative is another story, but no, none of you are screwed, you will be able to exchange your phone.
Target is being cagey because they do not know if they will sell the Note 7 anymore, so most reps do not expect to get them in to have one to exchange for your old phone. I have been told that even if they are not selling the Notes7s they should have Note 7's for exchange. We'll find out tomorrow.
But my advice is just to chill. None of you will be stuck with your old Note 7.