Welcome to Wefugees and Thank you for sharing your question with us!
Am I understanding you correctly - your first and follow-up application for asylum were both rejected?
If that is the case, you are technically obliged to leave the country ("ausreisepflichtig"). As the following source states, you have the chance to appeal the rejection of your follow-up application. In order to avoid imminent deportation, a so-called "Eilantrag"/ urgent appeal has to be filed with the appeal against the rejection. It's purpose is to receive a court order that discontinues the deportation until a decision on your appeal has been made.
Source (in German, please use a translation tool): https://www.nds-fluerat.org/leitfaden/4-der-bescheid-des-bundesamtes/42-kein-neues-asylverfahren-nach-dem-folgeantrag/
Coming back to your question: Filing an asylum application for your children gives you (as the parents) the chance to receive the same protection title as them once they receive a positive decision. Unfortunately, I can't predict the outcome of this case. You can find more information on family asylum here by the BAMF: http://www.bamf.de/EN/Fluechtlingsschutz/FamilienasylFamiliennachzug/familienasyl-familiennachzug-node.html
Please note that I'm not a lawyer, so the information I shared here stems from my own research. If you already have a lawyer, he will be able to assess your chances much better than we are able to here! Other than that, there are refugee councils and legal aid services as well that can have a look at your case.
These are the pro bono legal aid services in Rheinland-Pfalz: http://www.refugeelegalaidinformation.org/rheinland-pfalz-pro-bono-directory
And here is contact information of Caritas Kaiserslautern: https://www.caritas-zentrum-kaiserslautern.de/ich-suche-hilfe/migration-und-integration/migrationsberatung/migrationsberatung
I hope this helps. Feel free to contact us again if you have any more questions!
All the best,