Which state's inheritance tax laws do we follow?
You do not have to pay state inheritance tax or state income tax. Your mother lived in Ohio, so that state determines whether any inheritance tax must be paid. Under Ohio law, her estate was not subject to inheritance tax. Tennessee law, that state of your residence, has no application to your mother's estate.
You do not have to pay income tax (federal or state) on your inheritance. Inheritances are not subject to income tax. [Of course, there are exceptions, such as money left in an IRA that was not taxed before.] You have inherited the money free and clear.