Unless your current country allows dual citizenship, you CANNOT travel to abroad without a US passport AFTER becoming a US citizen. It's against the US law, and I'm sure it would be against the law of your current country too if someone who is not their citizen would use a travel document (passport) issued by that country. Don't forget that you won't be a citizen of your country after becoming a US citizen unless your country allows dual citizenship. Plus, you cannot enter into the US without a US passport anyway if you are a US citizen. Naturalization certificate alone is not enough. Thus, you cannot use your previous passport issued from that country to travel in and out of US.
You have only two choices- The first one is to make an early appointment with Passport office regardless of where that office is located. If they are located so far that you would need to take a plane ride, then so be it. They will be able to issue you a passport right there and then within a few hours. And if they would give you a hard time to issue you the passport on the same day then speak to their supervisor who will be able to accomodate your request without a problem. However, make sure to take an airline ticket and some kind of proof to show the emergency to depart the US at very next day, in case if they want to see it to justify the expedited request.
The second choice is to delay/postpone your travel which you would be doing if first option won't work. There is no ifs, ands, buts, would, should, could...
Good luck.