I do not know what a dcas disqualification for character means, but if you were convicted of V&T misdemeanors, they are unclassified misdemeanors, but crimes nonetheless. Usually, mere traffic violations are ignored unless germaine to the employment sought to be filled. So while any associated points may have expired, the conviction will still remain. Computers never forget.
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You should appeal your disqualification if it was solely on the basis of those convictions. New York has a law that prohibits both state agencies and private employers from denying someone employment on the basis of that person's prior criminal record unless the crimes that person was found guilty of either (a) bear a direct relationship to the specific duties of the job in question, or (b) employing that person would pose an unreasonable risk to public safety. See NY Corrections Law Article 23-A.
There is no guarantee that you will be successful in having this determination overturned, but it is worth appealing unless they stated other legitimate reasons for not hiring you that weren't related to your criminal record.
You should also immediately consult with a competent employment/labor law attorney because you have only a very short period of time to contest the DCAS determination (approximately 4 months).
Good luck, and stick up for yourself; the convictions you described don't sound - by themselves - like legitimate reasons for denying you a job unless the job has to do with being a security officer or someone who handles money.
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I agree with my colleagues. You should contact a competent attorney immediately because of the very short statute of limitations described in the other answers to this question.
The author of this posting is licensed to practice law in the State of New York. He specializes in litigation matters relating to personal injury, construction accidents, auto accidents, slip and fall, dog bite, contract litigation, property litigation, civil rights, ERISA, and Social Security matters in federal, state and local courts, with a focus on courts in Staten Island, Manhattan, Brooklyn, Queens and the Bronx. This posting is intended as general information only, is not provided as legal advice in connection with any specific case, and should not be construed to create an attorney-client relationship.