I think fescue might work. Fescue does very well in the shade, but I am not 100% sure if it is ok that far north, I’m sure if you google it, it will tell you where it grows. Lowes and home depot sell grass for shady areas, and then another one for VERY shady areas (they are both a mix of a variety of fescue for the most part). I usually buy the two and mix them together, less of the very shady because it is very expensive. Best time to lay the seed is fall, second best spring when it is not too hot. If it is 80 degrees or higher during the day I think the seed might not take. Also, fescue does not grow in, it grows where you put it, and I think if it dies its dead. Other grasses if they don’t get enough water or turn brown will come back, or make babies and fill in some minor bare/thin patches on its own, not shady grass.
I’m no expert, but this has been my experience, I have a lot of shade on my lawn, and it is always a littel bit of a battle to keep it green, becuase I live in the Memphis HOT. I think you will have a musch easier time with Ohio being a little more moderate.