Welcome to the Wefugees Community and thank you for reaching out to us.
From my perspective, your situation doesn't sound too complicated actually.
First of all, I am happy to hear that the relocation worked out - sometimes it is really difficult to get a permit to move during the asylum procedure or even in the aftermath. Secondly, it is totally normal that you haven't received an answer from the BAMF yet after one month. As long as your asylum case is pending, the immigration office is issuing a so-called "Aufenthaltsgestattung" (I assume it is the document you are holding now which is valid for 6 months). It is also possible that the BAMF will not finance a so-called "integration course" including language classes yet. But they will ask you to get enrolled as soon as the BAMF acknowledges you as refugee or grants the Subsidiary Protection. If you spend a certain time in Germany, the immigration office issues a work permit (even during a pending asylum procedure).
As you are also entitled to receive social benefits ("Asylbewerberleistungen"), you should be able to survive even if you do not work fulltime. Maybe it is the more sustainable way to focus on the language and work maybe part-time (Minijob, for instance)? But at the end it is totally up to you.
If you want to discuss your options more detailed and on an individual basis, you may also contact a migration counselling office or an expert via the Mbeon-App.
I hope this helps and don't hesitate to get back to us with any further questions.
All the best,