I've never had such a case before. You can anyway stay in Germany at least 8 weeks after the birth, in this time the Ausländerbehörde can't insist on leaving Germany, it's a period of mother and baby protection (Mutterschutz). Naturally you must inform them about the birth and, if your free 90 days period is over soon, ask for a prolongation. In this time you can try to make documents for travelling. If you don't manage it in this time, you can ask the Ausländerbehörde for more prolongation, explaining how much time you probably need.
You can make a birth certificate for the baby in the Standesamt in the place the child is born. You need the parents' passports, birth certificates, I presume a marriage certificate too, And the registration of your husband in the place he lives. Contact the Standesamt on phone, ask what they need exactly, and make an appointment. With this document you can make a passport for the child in your embassy. I don't know if it is possible just to include the child in the passports of the parents; anyway the own passport is better.
After the § 33.1 Alien Law (Aufenthaltsgesetz) it must be possible for the Ausländerbehörde to give your baby a residence permit, but they must not do it, they have a discretionary power, because only one parent has a residence permit. I would anyway make an application. If you already have a permission for a family reunion and just have to pick up your visum (as I understood from your words), that means, the Ausländerbehörde has already proved the situation and is sure that the residence of your husband in Germany is of long-term nature and financially secured. And they know that you will also get a residence permit as soon as you get your visum.
Having a passport for the baby is a condition of issuing a residence permit for him/her. Passport is a general condition for all residence permits except in case of refugees.