Moving out after the year lease is up

Unless there is a written agreement that provides otherwise, your landlord is correct. When a written lease ends and is not renewed, the lease converts to a periodic agreement based upon the manner in which rent is to be paid. For example, if you pay monthly to rent the premises, then the lease becomes a month-to-month lease based upon an oral agreement. Essentially, you and the landlord enter into a new lease agreement every month that you stayed and are obligated to pay rent for that entire month.
Under Ohio law, a month-to-month lease can be terminated by either party with at least a full term’s notice of intent not to renew the lease or to vacate the premises. That means that you must give at least one month’s full notice of intent to move out. And the notice must be given before or at the start of the term. If it is given after the start of the term, then the 30 day period does not begin to run until the start of the next term. For example, you have a month to month tenancy. You decide on April 15 you are going to move out. You give notice on April 15th of this intent to move out on April 25. The 30 day period would not begin to run until the start of the next term, which would be May 1. Therefore, in this example, you would be responsible for all of April’s rent, as well as all of May’s rent, since the notice period does not begin to run until May 1, and the 30 day period does not end until the end of May. Unfortunately, Ohio law does not provide for pro-ration of rent when you move out in the middle of a term, unless a written agreement states otherwise.
Best of luck to you.