You have so many wonderful wild edible woodland plants over there in Eastern North America, some of which are really quite high yielding. I was blown away by a visit to Vermont and Eastern Canada last September. It was almost a religious experience to see so many of the woodland edibles that I've grown for years in my garden in Norway for the first time in their natural habitat, muixed in with various European invaders like Hedge Garlica (Alliaria) and Burdock (Arctium spp.). I'm sure many also grow in Ohio, but what to plant would depend very much on the type of woods, whether the woods have sufficient moisture in spring etc...
Some of the favourites I saw in Vermont are WoodNettle (Laportea), Waterleaf (Hydrophyllum), Sweet Cicely (Osmorhiza), Honewort (Cryptotaenia), False solomon's seal (Maianthemum racemosum, although the western form has reputedly a better taste), Hog Peanut (Amphicarpaea), Ramps (Allium tricoccum), Ostrich Fern (Matteuccia) etc. . Heracleum maximum also tolerates quite a lot of shade.
I have a page on my website about my trip and visit to the distributor of my book Around the World in 80 plants, Chelsea Green) with links to several albums with pictures of edibles I found:
That's just the native plants....there are also many other non-native perennial woodlanders you might like to try like various Hostas and Hablitzia tamnoides (Caucasian Spinach), although your summers may be too hot and dry for the latter. Similarly, Allium victorialis, a large broad-leaved onion. My book has a look up table at the back where I indicate good candidates for the forest garden!