I'm processing a new passport but don't know if I will have it in time to process the visa.

Fernando, you should apply with your new passport. It usually takes four working days to apply for Chinese entry permit. If you don't have time, you may choose the rush service. It takes one working day.

It's true that some airport immigration officers may let you in if you hold two passports. However, once your old passport becomes invalid, the entry permit on it automatically becomes void too, according to related rule. So it is too risky if you choose to use two passports to enter.