What you want to do is file as a non-resident Ohio return, paying tax on your Ohio source income. You file a NY full year resident return and pay tax on all your income. NY will give you a credit or partial credit for what you paid Ohio.
Whether the Ohio tax authorities will consider you to really be an Ohio resident remains to be seen. If you maintain a permanent NY address, the NY tax authority does consider you to be a NY resident. You could be subjected to double state taxation based on this living arrangement.
If the Ohio cops notice you driving to work every day, with NY plates, they will stop you and demand you change your plates and drivers license. Ridiculous? Actually happened to my partner. But then she's cute as heck!
