My only real option at this time, given I don't have anything already lined up, is to accept the demotion and keep looking. Like they say, it's easier to get a job if you already have one.
The part that worries me is that they even offered me the alternative of resigning. Makes it sound like they don't care if I stay or go. On the other hand they have been known to fire people outright when their jobs became expendable, but they offered to keep me on even in a demoted role (this is not precedented with them, as far as I recall). If I stay, they would have to know that I would leave at the first opportunity. I just don't know -- I'm really confused.
