Generally, since Ohio is not a community property state, you would not have liability for your spouse's debt, but in the situation you described you would have liability to the hospital since you actually received payment as the policyholder. What your spouse did with the check, or what she should have done, etc doesn't matter. You might have a claim against her for reimbursement. I would suggest consulting with a local civil litigation, or divorce, lawyer for further assistance.
Being or not being a community property state has nothing to do with you being responsible for a medical debts. Why? if you were legally married at the time the "little woman" went for medical treatment, in most states, your marriage vows made you financially responsible for paying her medical debt, whether you agreed or not.
Remember, those vows said "for better, for worse, in sickness & in health?" That is usually interpreted to mean spouses pay for each others medical debts as long as the marriage is legally binding, even to 1 day before the divorce.
Hope this perspective helps!