As an F-Visa is a business visa it is illegal for any school to employ you with one of these. It is equally illegal for you to accept such a job on that basis. If caught you can expect at best to be invited to leave the country pretty quickly and you may be blacklisted against further entry too.

If the school offering you the job can only support an 'F' Visa it almost certainly means that they cannot get you a Residence Visa. (Z-Visas are now only used for bringing people in from outside China) They are almost certainly not authorised to directly employ foreign teachers. In the case of any trouble you will have no rights whatsoever. Therefore Beware!

When your present contract expires and it is not to be renewed, your present employer is required to cancel your visa and you are obliged to co-operate with him in doing this. Obtain from him a Letter of Recommendation (in effect a Release Letter) Once your visa is cancelled you have a certain amount of time in which to transfer to a new employer who can provide you with a residence visa. Or leave the country altogether.

My advice is not to accept the new job unless they can employ you in a legal manner and an 'F' Visa is NOT legal in your situation. People will tell you that it will be ok and that it is done all the time. This may be true to some extent - but it is an illegal situation and in the event you are caught you will be in trouble of some kind. If you do this you ARE putting yourself potentially in harm's way!

John