I didn't realize you were from NYS.
Nobody's going to care about the traffic infractions. The only thing you really have to worry about is the DWI, which is a misdemeanor
Whether this will turn up will depend on how deep a background check gets done and how many years they want to go back. If they don't go back more than 7 years, you will be okay.
I can't predict what will happen to you with this job, but why not apply for a certificate of relief from Civil Disabiilties? New York doesn't expunge cases and a governor's pardon here is rarer than a hen's tooth. Because of that, they have this certificate and if you get one, no NYS employer can discriminate against you solely because of your prior conviction.
You get it through the Department of Parole
, and it's a pretty useful document. I don't know whether you could get one in time to be of any assistance in your present job if your offense comes to light. But you'd have it in place thereafter. It's free , you can apply for it without a lawyer, and though it can't erase your crime, it can do you no harm. It might even give you a basis for a suit if you wrongfully don't get a position. You can find out how to get the certificate and what it does and doesn't do here.