Thank you for reaching out to us and I'm very sorry for the wait due to the Holidays.
When you received the rejection, did you appeal the decision? Did you receive a Duldung?
There are several threads on our platform that show options for asylum seekers who's claim has been rejected, here are some:
My appeal is rejected by the court- what can I do?
Also, here is an information site by Welcome BW that shows your options after you've been rejected - please follow this link. It explains the types of rejection and forms of appeal, perspectives in Germany after the rejection, Duldung, and more.
Regarding the status of your children, I found this paragraph on the BAMF website on family reunion:
"Born in Germany
When a child is born in Germany after the parents have filed an asylum application, under certain conditions, the legislature provides the possibility of a separate asylum procedure in order to protect the child. For the latter, the parents, at least one of which must still be undergoing the asylum procedure, or the immigration authority, inform the Federal Office of the birth. The asylum application is thus automatically regarded as having been filed in the interest of the newborn child. The parents can put forward their own reasons for asylum for their child. If they do not do so, the parents' reasons apply. Here too, appeals are available should the Federal Office reject the application.
Should the application be turned down, minor children will not be and may not be returned separately from their parents, also in the interest of child protection."
My sources state that parents of children who have been granted asylum/ refugee protection can be entitled to the same protection as their children, especially if they are minors and the parents have custody of them.
Just to make sure you know your rights in your case, I would advise you to consult a lawyer or a legal aid service for refugees.
Here is a list by ProAsyl of local refugee support services. Please select the area you live in: https://www.proasyl.de/en/our-network/
Of course you can reach out to the community again, if you have any further questions.
All the best and Good Luck,