Welcome to the Wefugees Community and thank you for your question.
During the asylum procedure you can unfortunately not apply for another residence permit. I would recommend you to discuss your situation in a migration counseling office, for instance. Depending on factors as your chances in the asylum procedure, you may want to consider to take the asylum application back, in order to apply for a residence permit according to i.e. par. 28 of the German Residence Act.
I hope this helps and please let us know if you need support in finding such an office or if you have any further questions.
All the best,