Please understand that the answers provided here are short and somewhat generic. Your lawyer probably told you that you can appeal the hearing officer's decision to a district court judge. You have a very limited time period in which to make such an appeal. The appeal to the district court judge is different than an appeal to the court of appeals. Whether your lawyer helped you or not is not necessarily dependent upon whether you won your point or not. But, that is why there is an appeal process. Perhaps the hearing officer did not sufficiently understand some aspect of your case.
The other part of your question has to do with settlement of a judgment. It sounds like the mother was on public assistance and the state was offering a way for you to 'buy down' the judgment. Settlement offers on judgments are (in a way) always open. Perhaps after you get your tax return, if you get a tax return, you can talk with the state about using that in some combination of trying to reach a settlement of the arrears judgement.
I hope this helps. Good luck.
Law Guru