Boy, I dont know about this time of year. Especially for just overnight. I checked one in Brunswick and they arent open yet. There is a state park called West Branch which is in Ravenna, but that would be about 50 miles southeast of downtown Cleveland. Their phone number is 330-296-3239. They told me today there is some road construction around there and you need to enter from ST. RT 5. The only have potable water avail at the dump station until this weekend. The bathhouses are closed until then. I wonder if you could camp overnight at the Camping World lot that is located at Rt303 and I71. I'm thinking that is about 15miles from downtown. Their phone number is 800-528-4566.
Sorry I couldnt help more.
Mike And Debbie- Northeast Ohio
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