Welcome to the Wefugees Community and thank you for reaching out to us.
First of all, congratulations to your unlimited work contract! That is already an achievement! You are also working more than 15 hours per week, right?
From my perspective, there is no need to hire a lawyer in the fist place. I would recommend you to reach out to a counselling office to get support in your cause. Together you may formulate the "Umverteilungsantrag" (application to move to another city/district) and point out your arguments and reasons. You may look for an apartment already, but I would advise you not to sign a contract yet. I am sure the counselling centre can give you more individual and regional specific information. Please let us know if you need help in finding respective contact details.
I will link two of our experts here, too. Dear @mbeon-ruth or @mbeon-fardeen, can you add some information here?
In any case, please also don't hesitate to get back to us with any further questions.
All the best,