This is probably not as serious as you think. It's most likely just an administrative mixup.

The first thing to do is **tell US immigration when you left**. Don't worry about proof at this stage. Contact them back immediately and give them the date and flight number on which you left, and tell them where you have been since then. Make it clear you haven't returned to the US since. They will almost certainly be able to verify that you were on that flight, and that will probably be the end of the matter.

Also remember that, even if for some reason they don't believe that you left on the flight you did, you don't need to prove you left on that flight, you only need to prove you left before the end of your legal stay, ten days from now. The easiest way to do that is probably to make some sort of visit to someone official, like a government office or the police or a lawyer, where your visit will be recorded and you can provide information. But something as simple as a record of an everyday transaction will help - a few records of you using your credit card in your hometown is going to be evidence, and is probably good enough.