Under the common law, a step-parent had no obligation to support a step-child.The Ohio Revised Code does not have a provision for including the income of a step-parent in calculating the child support owed by the biological parent-spouse for children from other relationships. There is some Ohio case law that if a step-parent places him/herself in the role of "in loco parentis" (in the place of the parent) then they may be assessed for child support, but that requires overt acts by the step-parent indicating that they wish to take the place of the biological parent, and I doubt that you have undertaken those acts.
On the other hand there are now approximately 20 states that impose a statutory duty on a step-parent to contribute to the support of step-children during the marriage, but these vary as to the extent of the responsibility, and most do not speak to step-parent's obligation for the payment of child support to the former spouse. It may be argued that making a step-parent liable for the financial obligations of the new spouse is a detriment to marriage, as it would act to dissuade those considering marriage from entering into it if the non-biological parent would become responsible for the biological parent's child support obligations.
Since the Ohio Legislature has not seen fit to state its intent in law to make step-parents liable for the child support obligations of the biological parent, you should not be liable for the responsibilities of your husband.
Your husband needs to file a motion to modify his support obligation during the period of time that he is out of work. I also suggest that he keep a list of the places at which he applies for employment to present to the judge and show that he is trying to find work so that he may resume his regular obligations.
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