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You've been told right that the government will come after him for child support if you seek assistance.
Also, he can't get around paying child support as it is required by state law - neither of you can waive child support because it is money for the children, not either of you. Thus, there's no way to get around it.
The only thing you can do is ask the court to assess his child support for as little as possible - the court has some discretion with this depending on custody, how much time each parent spends with the children, etc.
Maybe the judge can assess his support in an amount that he can afford and that would allow you to be eligible for food stamps, child care assistance, etc.