Welcome to our community and thank you for your question!
Congratulations on your successful marriage first of all :)
I think what your lawyer means is to withdraw your asylum procedure and then to apply for a residence permit. You don't need the asylum procedure any longer to receive a residence permit because you have a right to get it through your marriage with an EU-citizen. Withdrawing the application doesn't cost you anything. But you will be charged a fee when applying for a residence permit. I don't know how much it is exactly as it depends on your place of residence and the legal-framework of the federal state that you live in.
I don't think you necessarily need a lawyer to apply for a residence permit in general, but since you've been rejected by BAMF already, I think it's good to have legal-support throughout the process. I'm not sure whether it's a good idea to withdraw your asylum application first and then apply for the residence permit. The Aufenthaltsgestattung that is valid for the duration of your asylum procedure legalises your stay. If you end the procedure you don't have this document that allows you to stay in Germany anymore. And it can take some time until you receive your residence permit instead.
If I were you, I would get back in contact with your lawyer and get precise advice on how to proceed. In my opinion it makes sense to let the appeal go on and to apply for residence permit at Ausländerbehörde in the meantime. Both procedures can run at the same time and it will be safer for you because you can't get deported until a final decision has been made on your asylum case. Hopefully you will receive a residence permit soon - then you can still withdraw your appeal.