Drunk driving convictions are in fact criminal convictions. They are misdemeanors, however, their statute citation is usually under the motor vehicle code; nevertheless, a misdemeanor a criminal conviction. Additionally, Reckless driving convictions are generally also criminal misdemeanor convictions as well; they likewise, are usually under the motor vehicle code, but are still a criminal conviction. Regarding your last question, "pending misdemeanors" are indeed "pending criminal cases"; all misdemeanors are criminal cases and/or criminal convictions. This is a long shot, but if your attorney can get your case knocked down to "careless driving" this is generally considered a civil infraction.
While considered a traffic violation in NJ, it appears to be a criminal offense in Ohio (misdemeanor) -
Ohio Revised Code, Title 45:
Chapter 4511.19 Operating vehicle under the influence of alcohol or drugs - OVI
(G)(1)(a) Except as otherwise provided in division (G)(1)(b), (c), (d), or (e) of this section, the offender is guilty of a misdemeanor of the first degree… [other sections pertain to repeat offenders].
You may want to check with another Ohio attorney for a definitive answer.
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To make matters more complicated, New Jersey does not have a "misdemeanor" category. Offenses are either felonies, and considered "crimes," or disorderly persons offenses, which by definition are not considered crimes in New Jersey. Motor vehicle offenses, such as a DUI, are neither.
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The problem is you are bound by ohio law. Does Ohio grade misdemeanors as a criminal charge. In NJ it is not but Ohio governs this one.