Brandon,
As others have noted, Shawnee is challenging, at best. Probably better for turkey than deer. It's full of narrow ridges and valleys, so lots of up and down climbing. Good news is you won't be overrun by a lot of other hunters.
One other observation about Ohio public hunting lands south of I-70. You will find a number of them, like Zaleski State Forest, that have few deer compared to 20 years ago. Several factors from my perspective. Private land is being taken up with hunting leases. In turn, the pressure on public hunting land is much higher. Zaleski has seen some extensive logging that has disrupted what used to be pristine forest and fringe areas. What started out as small plots, 40-50 acres appears to have grown to more than 100. Clear cut areas have not been maintained, etc.
Net, be sure to check out possible areas 2-3 times in different seasons before making a decision.