Technically, you are required to file a part year resident return for both NY & OH. But, if you had no income, during your short time in OH, you will owe no OH tax and for simplicity could consider just fling a full year NY return. You were not an OH resident, when you bought the house. You became a resident when you physically (and permanently) moved.
Also be aware that OH has both City and School District (SD) Income taxes. City taxes are only on earned income, but most SDs tax all income ( a few taxed only earned income).
To verify or find out if you live in a school district or city with an income tax enter your address at: