Can't you exchange your flight and depart a couple days later?
Also I'm curious to know what type of issues you had? Was the expiration date too close?

And this is why we don’t use our passports as dog chew toys!

Ask the embassy if they can call the airline for you. They may be able to work something out.

I would see if you can pay extra to get your passport faster, and see if you can get the embassy to call the airline. Or call the airline your self and what they can do to help you I doubt this is the first something like this has happened worst case is you might have to pay some more money to get on a different flight.

What country do you live in? If you're going to an embassy doesn't that mean you are already abroad? If in the US and a US citizen, there are more places than the embassy to get a new passport. If you show imminent departure tickets and are coolly panicking they will expedite it. Although I'm not sure how the gov't shutdown will be affecting them right now.

eek! But all may not be lost; often when they say "X" they mean "up to X", so it could - possibly - arrive in time. Or talk to the airline about going on the standby list.

Show up at the embassy every single day—to the point where they’re sick of seeing you. I bet you get a passport before 7 days.