Technically as a US citizen you are supposed to exit and enter the US on your US passport. However this is rarely checked, almost never enforced, and never punished – especially on exit where no passport checks are done.
It is perfectly legal to mail a passport. US passport service will not mail it overseas, so you will need to get someone to collect it from your home address and send it to you. Given the nature of the document I recommend a courier requiring a signature and recorded delivery.
Your other option is to call the passport service and see if they will convert your application to “expedited”, so it will be delivered before you leave. It involves a fee payment, but it is probably less than couriering the passport, and gives you peace of mind.