The Z-visa is only necessary for entering China legally for the purpose of earning income. Once you enter the country, you have 30 days to convert that Z-visa into a residence permit, which you have already done. There is no reason why your current employer would need to apply for a new Z-visa as—based on what you’ve written—your old employer did not close out your residence permit and downgrade you to a tourist visa. You still have a valid residence permit (based on the original Z-visa) and are earning income legally in China.
I believe the delay is a money-saving strategy only. Your FAO will probably wait until April 1 before taking your passport to the local PSB. She will then apply for an 11-month residence permit, as the schools save a lot of money by not asking for the full year. This approach proves to be inconvenient for the foreign teacher as, every year, the residence permit’s renewal date gets pushed up another month but this approach has become normal operating procedure across China's universities.