I'm really sorry that you are in such a difficult situation, in spite of all your efforts.
Unfortunately the problem is really difficult to solve. Normally, you only can sign a contract for a new living place (house, flat...) after the Ausländerbehörde's given you their permission. First you have to bring them an offer without signing it and ask for permission. Naturally it is a problem because the owners want to make contracts at once and don't want to wait. But there are so many problems in Germany which are just a consequence of burocratic procedures.
Especially if it is about changing the federal country (Bundesland), it is always very difficult. And I presume, it is the case here.
A big problem is that the Ausländerbehörde in the place to which you move must confirm that they agree to accept you. First, these requests (from Ausländerbehörde in your original place to another one in the place you aim to move) can take much time. Second, to get this agreement can be very difficult. They tend to reject.
It could be very helpful, if the company where your husband works supports him. They can address the both Ausländerbehörden asking them to give you a permission to move, explaining how good he is and that they need him and don't want to lose him because of this situation. Sometimes companies (bosses and colleguaes) are really very helpful.
You can also ask for help in the university. They have a foreigners department, a student organisation or perhaps more than one (I would definitely look what ASTA can do for you) and I think they must have something like "Antidiscriminieringsstelle".
It is also possible to write to BAMF about your situation and ask for a quicker decision. But it only makes sense if you count with a positive decision (for example if you are from Syria or if you are sure to have very strong reasons. Because if you make your rejection come quicker, it wouldn't help you with your moving but would only shorten your safe time for integration through work and studing during the running asylum procedure. So it is important to evaluate your chances.
From legal point of view: I think it must be possible to make an urgent appeal to the court (Eilantrag) because your case must be decided quickly, or you risk to lose the job and the university, because you can't pay for two living places and basically you can't stay in your new place without permission.
An urgent appeal can be made after rejection of your Widerspruch. If you still have no decision from the Ausländerbehörde, you can ask them in a written form to give you a decision within 2 weeks and if they don't, you can go to the court because your need to solve the problem is urgent. Naturally it is better to have a good lawyer for that. As I understood, you have a lawyer and perhaps he or she has already started this procedure.