Ask an Immigration Solicitor. Get an Answer ASAP.
I would strongly advise against overstaying your visa. It is a criminal offence and it will have a cascading effect on any further visa applications you make in the UK and probably elsewhere.
There is no solution except if you get an offer from the University you can apply to change to a Tier 2 visa. You must either apply for a Tier 2 visa or you must leave before the expiry of your visa.