Can I be successful in court terminating my spousal support because my ex is cohabiting?

And therein is the problem. Merely sharing the same domicile is not enough to prove co-habitation in Ohio. There are three generally accepted factors: the people in question must be living together, for a sustained period of time and are sharing expenses. You may be able to get their financial records in your search to prove co-habitation. You should discuss this with an attorney whose primary focus is domestic relations law.