My USA Passport expires in April 2017 and I would like to renew but can't send my current passport due to imminent travel. Is it possible to keep you old passport while your new passport is being processed? Can I apply at a passport facility?
Yes, you can keep your old passport while your new passport is being processed, provided that you have confirmed travel.
Per the USEmbassy.gov website:
Second passports may be issued if:
A passport is needed both for confirmed travel and submission for foreign visa applications simultaneously;
The primary passport is held by the U.K. government or foreign embassy and the applicant has confirmed travel; and/or
The applicant has confirmed travel plans to a foreign country which will deny a visa or entry to the bearer of a passport containing markings or visas showing travel to certain other countries.
The quote mentions the UK government because it's for US citizens who are in the UK. But in principle, it's the same: if your passport is being held by the government, you can get a second passport so long as you have confirmed travel.
I think the bigger issue is not just where and how to apply, which you can expedite through State Department through various means but also whether or not it is possible to hold 2 valid US passports at the same time.
The only mention I found was in a Blog that says that you can apply for a second US passport if your travel falls within a window of you applying for a visa from a 3rd country.