So you're leaving the country on short notice. Maybe you quickly bought a plane ticket for a vacation. Then you find out it takes a month to get a passport! DOH! Let's do this.
First things first. Get all the things I mentioned above as quickly as you can.
To get the application for the passport, fill out the form (https://pptform.state.gov/) Make sure you print two copies and don't actually submit it online. One copy will get sealed and the other will probably go to the expediting company.
After you get them you must select a company to help you expedite your passport. I think these companies are located in DC so they can get the passport quickly in person. I recommend a company called "Passport Visas Express"
Bring all your belongings along with $25 (Must be cash) to the local clerk of court that accepts passports (Call to confirm). They will help you figure out what goes in the sealed envelope.
Send the sealed envelope overnight to the expediting company along with whatever else they require.
Call to confirm they have received it and that you can get the passport in the time you need it.
Wait for you passport in the mail. If everything went well you should have one.
