Welcome back to the Wefugees community and thank you for reaching out to us.
I would highly recommend you not to move without the official reply by the immigration office (Ausländerbehörde) and their confirmation of the „Umverteilungsantrag“. I know it can be very difficult to reach staff and often it takes too long until applicants receive an answer.
If you don’t reach them via phone keep sending reminding emails - maybe also to the supervisor of the department highlighting the importance to move for your current job. I think waiting for 4 weeks is unfortunately rather normal than an exception.
I really hope that you can work remotely for a few weeks more. Searching for an apartment may also take some time and you could start with looking for places for the time being.
As my colleague wrote, asking a counseling office for support is always a good idea as well. With this search engine you can find a close-by organisation.
Please don’t hesitate to keep us updated on this or to get back to us with any further questions.
All the best,