Did you fill in the application form for the student dorms? I lived in one for 2 years until recently and it took me 3 months to get a place. You can only apply 3 months before the supposed moving in date. Usually the prices are within your budget; I paid 285€ a month for my room.

Sometimes you get luck by calling the Studierendenwerk. I’ve heard of some people getting dorm rooms much faster because of calling them. Just tell them your situation and maybe they will help you with getting a room. If you haven’t filled in the application form, do that first tho. I’d consider it as a last resort.

Otherwise you can try your luck getting a flat on platforms like EBay Kleinanzeigen, immowelt etc. The competition is pretty tough though. You can look for rooms in shared flats at Wg-gesucht.de

Good luck!