Dear @Mikou21,
Welcome to the Wefugees Community and thank you for your question.
Every federal state in Germany has its own hardship commission and they work slightly different. Therefore, it is a bit difficult to give general information. I am more familiar with the hardship commission in Bavaria (Bayern), for instance. In Bavaria I would not recommend to send a general application to the hardship commission office (Geschäftsstelle) at the Bavarian Ministry of Interior. They would inform the members that they received an application but the chances are not very high that a member picks your case up in order to defend it in the commission (which would be a requirement) since the members receive many applications directly. Therefore, I would also always advise to contact a member of the hardship commission (here the contact details for the Bavarian commission which is only responsible for people living in Bavaria) instead to get a faster response and to increase the chances.
However, maybe we have to clarify some facts which should apply to all hardship commissions in Germany. The hardship commissions only decide on cases in which people are officially obliged to leave Germany and who do not have any other perspective/chance to stay in Germany. It is really a very last resort. Usually the hardship commission wants to see that you are well-integrated in Germany and a deportation is likely to put you in a challenging situation (see also Handbook Germany on this). If you speak German well enough, attend school, go through vocational training, study at university or have a job, you have a higher chance of being issued a residence permit. Furthermore, it is beneficial if you have some volunteer work experience and many friends and acquaintances (in Germany) who can support you in your hardship application.
Based on the information you share, however, you might be able to receive a residence permit as a family member of a German citizen, no? In any case, you really have to make sure that you tried any other possible way to get a residence permit in Germany before approaching the hardship commission. I highly recommend you to discuss your situation with a migration/refugee counselling office to ellaborate your individual case and the next steps to take.
I hope this helps and please don't hesitate to get back to us with any further questions as well.
All the best,
Meike