In order for them to "change my visa" wouldn't the previous one need to be cancelled so they could re-apply for the same document that says it's issued in Beijing? Is there really any reason they would do this while my current permit is still valid?
old34 said it better - but I spent so long tapping out a reply that I am posting it anyway
The problem is that your 'visa' (Residence Permit) is neither 'current' or 'valid' from the time you terminate a contract.
Your Residence Permit was issued against a Foreign Expert Certificate, which was derived from the Work Permit issued following an application made by your previous employer.
In other words, you were granted a Residence Permit on the strength of your having an approved job with a specified employer. Once you have finished working for that employer the Residence Permit is no longer valid.
While an employer is duty bound to notify the PSB of the contract's early termination not many do so. If, however, an employer is pissed off, and does notify the PSB, the Residence Permit will be cancelled.
A new set of documents are required before you can legally start work for another organisation - Work Permit/FEC/Residence Permit. The starting point for that is a Leaving Clearance document from your previous employer, which specifies the date of your termination.
I would be really concerned if the new employer is baulking at doing that; any good employer who has authority to employ a foreign expert would not take the risk of operating outside the law.
Without new documentation you are at risk. I have no doubt that if the PSB did check and find that you are working on a Residence Permit derived from another employer there is every chance that you would be asked to leave the country.
I think it is best if you ask the new employer to start making arrangements for your new documents once you have forwarded the Letter of Clearance from your old employer. You could point out that it protects both you and the company from any problems in the future.