Hello Salome Mkota,
I don't think you must be afraid of making the visa procedure, if you can get the Vorabzustimmung before. This is the idea of Vorabzustimmung: if a person provides such document to an embassy, the embassy staff doesn't have to ask the Ausländerbehörde anymore, they only check if you have all necessary documents in a proper form for issuing a visa. That means, you can prepare everything from Germany, check the set of documents, get your Vorabzustimmung, make an appointment with the embassy and buy your ticket for a date just before this appointment. You mustn't spend months in your country waiting for the appointment, you wait in Germany.
I've never heard that it doesn't work properly if you have a Vorabzustimmung.
The fact that you came illegally ist just the reason, why the Ausländerbehörde wants you to make this visa procedure now. The basis is §5.5.1 Residence Act (AufenthG): coming with a proper visa is one of the conditions for issuing a residence permit. The Ausländerbehörde can decide not to use this rule and give you your residence permit without leaving Germany, if they beleive in a hardship case. But it is their discretionary power and in your case they obviously don't beleive that it is fully impossible for you to make this visa procecure.
If you have never got an entry ban for Germany of Schengen area, your illegal entry can't be a problem. Ask the Ausländerbehörde if you have one to be absolutely sure.
It is important that you leave Germany legally, officially. Discuss with the Ausländerbehörde, I think they will give you a Grenzübertrittsbescheinigung which you can stamp in the airport or in the embassy in your country. They will explain you.
The alternative to this procedure can be a legal dispute with the Ausländerbehörde in the court and I am not sure that you win. As I told before, you will be in need to give proof that leaving Germany even for a short period of time is impossible for you. This can be difficult if there are flights and the embassy is working. And it will perhaps cost much money.
Naturally it is for you to decide. Speak once more with your lawyer.