Ok, I've got it all sorted out now. Thanks for the help!
There was a synthetic cork stuck in the chute. This is a brand new corker, so don't ask me why it was there. It was really stuck in there, but with enough force, it came out.
Once the cork was gone, it was immediately obvious how the corker is supposed to work. I just corked my bottles, and they went in no-problem.
Thanks again, everyone!