Get a letter from your school with the date of your last day of study and take it to Immigration and they will cancel your Extension of Stay and give you a new stamp up to the date in the letter.
They are the out-sourced agents for the UK government for Visa applications and passport renewals etc. You have no choice but to use them as when you start an application on a UK government website it will direct you to the VFS Global portal to continue the process. I've used then for a UK Visitor Visa.
That, at best, will get a Non-O single entry Visa giving 90 days stay so only two entries, the second on a re-entry permit. It seems like a multi-entry Tourist Visa might get the OP 3 or 4 entries if timed correctly if he is under 50 and cannot get a retirement Visa.