A&W website has a locator search engine, and it turns out there are still A&W's out there. Locations in Ohio include Kent, Tallmadge, Ravenna, East Liverpool and Orwell.
But if you'd like to experience the next best thing to a real vintage A&W Drive-in, why not take a nice drive down to Ashland and visit the Ashland-Wooster Drive-in? It was still a bona-fide old-time A&W Drive-in while I was in college, and every once in a while I would detour over to Ashland on the way home from Ohio State to get a chili dog for the last lap of the drive to Lorain.
According to their website, they've been there on US 250 since 1957, and were a franchised A&W until 1985 when the current owners took over. They still have carhop service (as well as drive-through service if you're in a hurry).
The drive down there is just a little over an hour.
And on the way home, don't forget a side-trip to Grandpa's Cheese Barn, as well as Fin, Feather & Fur Outfitters. They're both on US 250 as well, and only a few minutes from State Route 89, which you can take north to join up with State Route 58 again,
The photo of the Ashland-Wooster Drive-in above is from a flickr® site featuring great vintage sign photos, as well as other cool sets, by a photographer named Scott, who shot them all right here in Ohio. Click here to visit it!