You cannot legally teach for even one day until you get the Z visa, make this clear to the school
They should help you, but it is always your personal responsibility to get all necessary visas to visit, live or work in CHina, the school is NOT responsible.
After all, the school cannot be deported or fined, but you CAN.
Arrive weeks before school starts, so you have plenty of time to get the Z visa, physical exam, foreign experts certificate and resident permit... It is time consuming, but painless and cheap.

Paul Barry