Platesandscrews,
Join the January 10th club. Know that you are not alone. As someone who had to choose between multiple programs on that date alone and unfortunately turn down a couple, I'm acquainted with your pain. I'm certain there are many on this board that had to make similarly difficult choices, especially for that date. I do wish you luck in finding someone to swap with. However, consider that the longer you hold onto multiple invites, the longer you deprive other candidates of the opportunity for an interview.
OF