Very Confusing Visa Situation. Sorry for the Length.

I lost my passport (I am American) while in Beijing, and while it was lost I overstayed my visa. I have overstayed about a month and a half now, and I am prepared to pay all necessary fines. I have a flight booked on October 21st, but I am not sure how to pay my fine and exit. I have no physical visa in my passport, but I do have the photocopy of my old expired one. Will this be enough to get through the airport? If not....people then tell me to go to the PSB and get a new visa, but then...

The PSB tells me I need to register my new passport to get an exit visa, but I can't get register without a valid visa. I am lost and confused in this situation. I have looked all over the internet and talked with various agencies, but my situation seems to be unique and people can't help. I just want to pay my fines and leave China. Please help!