Put your significant other to work online acquiring a list of universities, colleges and top level high schools in the district of Beijing where you will live. Have her compile the FAO department email addresses from online data or by calling the schools. Tell her not to stop until she has more than 50.
With a brief cover letter in English and Chinese (translated by your girl), email your resume to each of the schools ASAP. You are applying late; but many schools are understaffed and desperate.
In the meantime get busy locally apostillizing your degree and acquiring your non-criminal record from the local police.
You will receive a handful of offers from the emails you send. Compare them.
If any recruiters contact you, tell them you would rather stick a sharpened #2 pencil in your eyeball than do business with a Chinese recruiter.
Do not get on a plane without a Z visa in hand.