thanky ou for addressing your situation here in the community.
The best way to move to another city is the job option.
Whether it is Vollzeit or Teilzeit it depends whether with a Teilzeit contract you will be able to cover your living costs and should be a at least 15 hours per week contract (§ 12a Abs. 5 Satz 1 Nr. 1 AufenthG).
As it seems that your case is still in process, the condition is that you have to be in Germany with a valid Aufenthaltsgestattung for at least 3 months, then you may move to another city. However, the Ausländerbehörde needs to agree to your move.
For the Application it is important that you have a specific job offer and you can go before the Ausweis Expire Date comes around.
I hope I could help you, if you have further questions please do not hesitate to share them with us here in the community.