Let us all know what happens please. This is first class ignorance... We would all love to know what happens to you! Good and bad please. I hope it all works out for you but please tell, haha!
Your only option from what you have said is to apply for a new passport. You will then have to pay an overstay fine. If what you are saying is correct you do not even have room for an exit stamp let alone an entry stamp to another country or any room for a TR visa. Pretty sure this is your only course of action. (Would love to know what airline allowed you to travel here in the first place?)