It is probably right that your lawyer didn’t want to make a further appeal to the higher court because it has no chance in the most cases and would just cost you a lot of money. And it is not surprising that the judge gave you a hope to stay and then your appeal was rejected. The administrative court (Verwaltungsgericht) decides only about your asylum reasons or the reasons for protection in Germany because of the danger of prosecution / unhuman treatment in your country of origin. If they find your asylum reasons insufficient or unbelievable, they don’t give you protection und make a negative decision independent from the fact that you have other reasons (in your case family reasons) to stay in Germany. It is the task of the Ausländerbehörde to decide about residence permit for all reasons other than humanitarian protection (asylum, refugee status, subsidiary protection, ban on deportation).
Normally you won’t keep your Aufenthaltsgestattung for 6 months, its validity is finished with the end of your court procedure. But it depends on your Ausländerbehörde how quick they take your Aufenthaltsgestattung and give you a Duldung.
It was really not possible for the Ausländerbehörde (Landratsamt) to decide about your residence permit for the aim of family reunion before your court procedure was over. It is a condition from the German Law (Aufenthaltsgesetz, §10).
There is no guarantee that you can get a residence permit because of your children without leaving Germany and making an application for family reunion in the German embassy. It is a “normal” procedure after the law and some of the Ausländerbehörden try to force people to do it. But in your case you have many circumstances which speak for the possibility to stay in Germany and get a legal status without leaving. On one side the humanitarian reasons like children’s interests and their right to have both parents and not to be separated from the father for a long time, which can happen if you must make a visum procedure. On the other side your good integration, your working place, earning your living, possessing identity documents etc. And if your Landratsamt has already said to you that the residence permit would be possible after the end of the court procedure, this can also be a good sign.
The Ausländerbehörde must make an official decision about your application and give it to you in a written form. If it is a refusal, they must give the reasons and an explanation how and within which time you can make an objection (Widerspruch). You can also fight it in the court. In case of refusal I would ask a lawyer for help.
And if you don’t achieve a positive result there is still a possibility to make an application to the Commission for Hardship Cases (Härtefallkomission) in your federal state (Bundesland).