Is there another way of obtaining a resident card if I don't qualify for deferred action?

I've been in US since age 6 illegally. I'm 35 now. I went to school here in TX. My mom is a legal resident and she petitioned for me in 1997 before I married. My husband is also, illegal. Is there any way I can hurry the application along? I also have a son who is 19 years old. I have a 9 year old daughter that has scoliosis and needs surgery every six months. My license has been expired for 2 years and at each doctors visit they need a valid or current ID. I've gone to Mexican consulate to get an ID but they can’t give it to me since my birth certificate and my expired license don't have the exact name. My license and school records only show my middle name and last name. I've never used my first name. I'm in desperate need of help. What can you suggest I do?