I am not an Ohio attorney but if this is anything like Virginia, DC, or MD, you file the Motion for Stay of Execution after the judgment has been entered.
Check your local appeal procedures, but appeal is probably a multi-step process with several deadlines longer than fourteen days. I don't think you would need to draft a full appeal and the motion for stay in just two weeks. Consult the Ohio rules and an Ohio attorney familiar with appeals.
Answers I write on AVVO are just to help you out, and should not be considered to create an attorney-client relationship, or even as true legal advice. In fact, most of my answers are just commonsense advice that I would give any friend, just with the benefit of my legal perspective. If you are interested in pursuing representation with me however, please contact me. My firm practices in Virginia, DC, and MD, both state and Federal courts.
You should have a motion for stay ready to go at the time of trial. It isn't absolutely necessary, but it increases your chances that the motion for stay will be granted.
If it is denied, you can re-file the motion to stay contemporaneously with your notice of appeal in the Court of Appeals.