I went through this process recently. If you can prove that you are leaving the country within 14 days then you can get it with just a little bit of hassle. You’ll need to call the 1800 number for the U.S. passport office, and then try and get an appointment at any office reasonably close to you. This is the hardest part. The first few times I called they claimed they only had appointments available in Puerto Rico but people cancel their appointment, so I kept calling and got an appointment about an hour and a half drive from me. YOU CANNOT MAKE AN APPOINTMENT ONLINE, YOU MUST CALL THE NUMBER. THEY WILL NOT LET YOU INTO THE OFFICE WITHOUT AN APPOINTMENT.
Once you get the appointment make sure that you have all your documents, including old passport, social, birth certificate, license, official passport photo and proof of departure, as well as the passport application which you can pick up in the post office I think. Make sure to get there early, as there is crazy security at these places. If your appointment is the same day as your flight they will do it on the spot, if not they will either mail it to you or you can come back and pick it up a day or two later. Don’t get it mailed as there’s almost no chance it will get to you on time; it’s annoying, but the best option is to drive back there, or they will give you a receipt that will allow someone else to pick it up for you, as long as you sign it or something, not totally sure because I just picked mine up myself. There will be some fee, not the cheapest but definitely way cheaper than hiring one of those agencies. Not sure if they accept cash but they do take care or checks as long as it’s in your name.
Hope this helps