Agreed with Piano, You need to consult an immigration attorney.
Now excuse me for being forward or blunt'
The fact that you entered illegally and fraudently gained employment and possibly other benefits by defrauding the system does not look good, it is grounds for deniel. And in my opinion you shouldnt be allowed any leniancy for doing this illegally. There are hundreds of people on here that enter legally, do not work and wait months even years for their EAD before they start working they have to make do with less, & pay thousands of dollars to attorneys,and still have to wait through the process. They are more deserving of GC & Govt. priviliages than someone who intentionally defrauds the system and illegally enters the country, I have no sympathy for your situation ,
However,for the sake of your children I hope you find the right answers and are guided towards the legal direction to gaining GC citizenship & LEGAL EMPLOYMENT!
You need to consult an immigration attorney.