I would narrow down a few areas to concentrate on. First, I got a map of Ohio. Then I drew a circle around the major population centers and guessed about how far someone would be willing to drive if they were making day trips. I ruled those areas out. That helped me to narrow down my search. Just like here at home, I look for out of the way spots that others overlook. I also planned my trip for during the week when work would keep a lot of people out of the woods.
If, (when), I go back to Ohio I plan to do my best to pick up some small, private, permission properties to hunt also. I'll use the same tactics I use around here. Identify an area I am interested in, look for promising spots through online mapping resources, determine ownership and contact the owners to see if they would be willing to let me hunt the property. This, like cold calling, has about a 4% success rate. That said, if you contact 100 landowners you can expect about 4 to say yes, on average. One great small private tract can be awesome. Don't be offended when people say no. That is going to be the norm. Expect it. Be polite, thank them for their time, wish them well, and move on. Also, if you do get a permission treat the property great. Abusing a landowner's property is the surest way to lose that permission. Once a landowner sees you are an honest, trustworthy person they may recommend you to their neighbors, friends, and family.
Networking is also a good way to get permissions. Think of everyone you know and if you know someone who lives in Ohio or travels there regularly you might contact them and ask their advice. They may live in a big city but someone they know may have a great small tract just waiting to be hunted.
If you are determined to hunt only public land I am sure you will be able to find some good spots. There are some giants roaming those hills.