This is a very common situation. The employer doesn't want the hassle of withholding tax for a single employee in another state, particularly a state like Ohio, with complex local tax, as well as state tax.
You need to accept the fact, that you need to make quarterly estimated payments to the Ohio Dept. of Taxation (ODT), for state income tax.
To find out if you live in a school district or city with an income tax enter your address at:
http://www.tax.ohio.gov/Individual/LocalTaxInformation.aspx
If you do, you'll need to make separate quarterly payments to the local taxing authority.
For a list of all Ohio cities with an income tax go to
https://tax.ohio.gov/business/municipalities/08municipal-income-tax-forms
From there, there is a link to each city’s web site and/or RITA / CCA (agencies that administer city income tax for many Ohio cities).
School District Estimated Payment Vouchers
https://www.taxformfinder.org/forms/2023/2023-ohio-form-sd-100es.pdf
