Does your permit card (the photo/Id one) list the name/details of the employer ?

That can be an indicator of whether it's 'open' or 'tied' but really the best answer will come from the Arbeitsamt.

Also, if you have been working for several years in Switzerland, then if your current employer fires you, you should be entitled to unemployment insurance, and in that case you can contact the RAV (unemployment office) - in fact, I've heard it said that you should contact them immediately as soon as you think your work situation is under threat - they will advise you on the procedure and what entitlements you have...

It should be fine...I am sure you can make a few phone calls on Monday and get good answers to your problem. If the relationship is good with your new employer, maybe they will do the phone calls for you (my new employer was more than happy to handle all the paperwork when I changed jobs, she was just so glad to have me on the team!)...