I hear your struggle. Suggestion if possible. Get rid of one of your cars and use that money to pay off bills. I am not sure where in NJ you live but if you are in a city you should be able to walk and get groceries, etc.
I believe in stay at home mom, we do the same but fortunately I do not have money problems now. Do you have any skills? Did you ever think of selling stuff on Ebay to help. Lots of home work ideas while taking care of the kids. I have three dogs and care about them alot. Is it possible to let someone care for your dogs until you get on your feet. These are only ideas, I hate to be a Monday morning quarterback.
Good luck with everything..