I think I found the location of the missing Hancock County marker listed as "Hancock County/State of Ohio". The location of this marker in the list is "First Street Weirton west of Carolina Avenue". I couldn't find a Carolina Avenue in Weirton.
The only logical pace for a State of Ohio marker to be located wold be near a bridge since Ohio and Hancock County are separated by the Ohio River. With that in mind I headed out Freedom Way toward the Fort Steuben Bridge. Sure enough, I found this evidence of a missing marker. That's the old Fort Steuben Bridge in the background. It is currently closed. I don't think there are any plans to restore and reopen it. I'll have to do more research on it when I get time.
I looked around a little bit to see if I could find the remnants of the marker on the ground but the crazy weeds were too thick.
Here's another shot of the pole with the new Route 22 Bridge in the background.