You need to contact a ticket attorney in your area to evaluate an appeal. It depends if the ticket court was a court of record. Witnesses can be present in the courtroom if neither party invoked the rule of procedure prohibiting their presence.
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We do TVB appeals. It's hard to tell exactly what happened from your description. Regardless, the transcript of the hearing will be the most accurate representation of everything that was said in the courtroom. Prior to reading the transcript, it's impossible to judge your chances of success. Generally, TVB appeals are quite difficult to win. You have 30 days from the date of conviction to file the appeal. Contact us if you'd like to discuss.
Generally most judges do exclude a witness during the time the Police Officer and the defendant testify. The fact that she was unsworn may act against you or may not depending on how it is presented. The appeal is done all in writing. No new evidence can be introduce. My office has been doing this for 21+ years with some success.
I'm assuming this happened at the Bronx DMV, where I work daily at. You make valid points in your questions regarding what appears to be improper procedure by the Administrative Law Judge. NYC dmv trials appear to be "court like" to motorist but unfortunately they aren't real courts in the traditional sense, they are "administrative courts." This means that evidence rules are not really followed slavishly by the administrative law judges. The other lawyers who have answered you are correct in stating that your chances of winning a paper appeal are slim. You have 30 days to file it from hearing date and it will cost you $10 plus a $50 transcript fee for a total of $60. In my 14 years of sending up appeals from the Bronx DMV, I estimate I have a 1% win ratio. I usually recommend that you are better off spending your $60 appeal fee on taking the 6 hour safety driving course rather than "gifting" more money to NY State. I don't do appeals unless I have fought the case but if you still want to appeal, please contact one of the other attorneys who say they are available.