You should consult a local DUI attorney, who should be able to answer your question quickly and during the free consultation, which most of us offer. In CA you would be able to get a wet reckless expunged once you qualified, and my hunch is you can probably do the same in OH. A quick call to a local attorney should help you out. Best of luck.
Ohio law does not permit traffic offenses to be expunged. These convictions go into your LEADS record and are also subject to the Interstate License Compact.
Ohio's expungement (sealing) statute was just amended last month by Senate Bill 337, and it afforded more rights to have multiple offenses expunged (actually sealed), but traffic offenses are still ineligible for this remedy.