How long can I stay in China after cancelling my Z visa?

Hi there,
First of all, thank you for all the amazing work you guys do. You guys do an amazing job at providing your readers with information on all things PRC.
Now for my question: I have been working for a school, through an agency, on a Z type for a year. Due to frequent violations of my work contract by the agency, I have decided to move on and not renew my contract with them.
After a long, emotionally draining battle, they have finally agreed to release me on the date that my contract ends. However, they would like to go to the PSB with me to cancel my work type on my last day at work.

Is it my work type they would like to cancel or my residence permit? If I agree to this, will the PSB give me any time to get myself organized to leave China? Can they give me a 30 day tourist type? Would you know how much the tourist type would cost? Also, is there any possibility that the agency could say anything untruthful about me that could result in the PSB not giving me another type in the future?
Please help.