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Joint custody is possible. You also may be able to watch the child during the days and that would go a long way toward minimizing your support obligation. There are a couple of cases that cna be used to put that argument together.
One question: if you work nights and watch the child during the day, when would you sleep and get things done (shopping, cooking, cleaning, laundry, paying bills, etc.)?
Anyway, there is a way out of this mess and you can probably avoid past due support with a well-crafted argument. You'll almost certainly need an attorney.
Questions? Call -- 312-987-9999 -- no charge, no obligation.
It is extremely unlikely any reduction in support would be achieved. Your financial situation is dire, but so is the situation of the children.
You have taken the DNA and it confirms the child is yours. I would recommend you preserve your limited funds and avoid litigation.
The author provides the preceding information as a service to the public. Author's response, as stated above, should not be considered legal advice. An initial attorney-client conference, based upon review of all relevant facts/documents, will be necessary to provide legal advice upon which the client should then rely.