There are many, many steps that have to be taken before your case went to trial.
And who knows, if your W accepts the grounds and doesn't contest the grounds and accepts your settlement offer all of this will be a non issue for you. I simply shared all this with you as something like 82% of all divorces in NY are contested. While mentally it bites that is also why 90% of ALL divorces are initiated via a Separation Agreement rather than divorce filings.
Before you went to trial you would have a RJI Hearing (Request for Judicial Intervention). Then several Pre-Trial Conferences. All of these proceedings take place in a little room (not a courtroom) and they are overseen by an officer of the court and not a judge.
There are MANY steps before it would go to trial.
My H was hellbent on getting a divorce and rushed, rushed,rushed the process (he did this to avoid being sued by me before I found out about OW but little did he know I knew!). He also told his attny that he and I BOTH wanted a divorce and we planned to use one attny, this was before I even knew he planned to leave. Once he saw I was not fooling around and our divorce case went to all the steps (this took 1.5 years) just short of trial he backed down and asked for a Separation Agreement. He knew he was screwed.
Professionally your W might simply agree to the grounds and your settlement offer as all of this will be public record. It would not bode well for her career to be sued for adultery for all her current and potential clients (patients?) to see.
That was my H's biggest fear - the info that would be on public record. His family had NO clue about OW until recently and the label of cheater via the legal system will never go away.
You have a LONG way to go before court. If you play it smart you won't ever go to trial but chances are you will have several pre-trial (RJI's and so on) to attend before *something* happens.