Emily - Do NOT let him talk you out of filing for support. I'm surprised you even have a choice - in my state, if you are on assistance, the state WILL go after the father for support - they want to recoup their expenses! I'm surprised your state is not mAKING you file for child support.
Feeding, clothing, housing those kids - he IS responsible for his share. Period. How much $ have you seen from him this year? And that would pay for what share of the costs of raising those kids? Yeah, I thought so.