Chinese laws/rules are suggestions which means that you should not expect a clarity. Moreover, forums are infested with multiple user names with twisted personalities that you need to be aware of. Depending on what kind of school or area you wish to come to you may or may not have to wait long. Private centers hire anytime while primary/secondary/tertiary schools with outlined semesters mostly in Summers/Winters. The gerenal understanding is that you ought to arrive on a Z visa that follows a formal letter of invitation; however, one of my collegues came on L visa which he was able to convert to Z in Hong Kong later. Anyhow, pay attention to what you are told by recruiters, employers and posters on the board carefully.

Over and Out
Caring

the boss has you by the balls once you have a work visa

If you are working on a tourist visa, the clowns at the agency slice your balls with scallions.

You received 'job offers' without the ability to provide a passport image or number?

I smell recruiters.

Employers in China require your passport information to apply for the documentation that must be presented (with your passport) to the Chinese embassy in your country of citizenship to acquire your Z visa. Without your passport information they are unable to provide these documents, which are tantamount to a 'job offer'.