Could you explain the process better? More specifically….

It‘s not as complicated as you thought. Both citizens of US and Canada have a chance to enjoy 72 hours free transit as long as they meet the requirements. Personally, I think you can enjoy 72 hours free transit. The rules have been adjusted. You needn't inform your airline for 72 hours free transit. When you land at the airport, just follow the signs to apply for free transit in person. If you stay at a hotel that receive foreigners, you don't worry about registration because the hotel does it for you. If you don't stay at a local's home, you needn't register at the local PSB too.

As for your dual citizenship, I think you should enter and exit with the same passport. Better not enter with Canadian passport and exit with your US one. If you are still worried about it, you might call Shanghai Airport Immigration Inspection at 0086-21-51105100 to confirm this.

Thanks for the quick reply. So I don't need to register with the police at all? I won't be staying with anybody. I just plan on visiting the city for a 4 or 5 hours and then returning to the airport. And of course I'lll only use one passport, but I will have both on me. I will most likely use my American passport but I reside in Canada.This may be outside your area of expertise, but will I have problems using my Canadian credit card in the city or to buy a train pass into the city? Just wondering if I should exchange some cash or if I can skip it.

If you don't accommodate outside of the airport, I don't think you have to register at the local PSB. I think it's better to exchange some RMB for buying tickets.