It is ILLEGAL to work in China without a Z-visa. You can be fined 500 rmb a day and permanently deported. The same is true for L and F visas.

The type of visa has NOTHING to do with the residence permit. That only shows you have registered with PSB.

Also Z visas can only be issued OUTSIDE of China. You would have to go to Hong Kong or some place else to change from a L or F.

Do not accept a job if they don't issue a Z visa before you arrive.