Can I sue this dentist?

Hi everyone, I’m looking for legal advice..

2 years ago, I saw a dentist that I believe was malpractice. I walked into the office with no pain at all. I made the appointment to get a cleaning. This is all covered by my insurance. Some minor details were that the receptionist wasn’t nice and seemed annoyed I was speaking to her. That’s besides the point of the actual damage that was done. I saw the dentist and he asked me if I was a new patient and I said yes. He replied with “I’m a new dentist” I thought it was just a dumb, meaningless joke.

Next, I had a cleaning by someone else that seemed normal. After, I saw the dentist. Again, I was in no pain at all and just there for a cleaning. He insisted my cavities needed refilled. He mentioned one molar. I agreed to him refilling it and thought I should trust him since he is a “doctor”. He then proceeded to drill in MULTIPLE teeth with NO anesthetic or numbing procedure. And also NO warnings or explanations beforehand or even after. I’ve had painful procedures done in the past and I’ve never complained. I had to let out a yell as I’ve never felt pain that like that before! It then hurt after. He told me it’s normal and the pain will go away and that my teeth and gums are just swollen. AGAIN, he did NOT numb me at all! So for a week and a half after it still hurt, so I went BACK to the same dentist a second time. (A friend told me to give the same dentist a second chance to fix something and if they can’t, never go back again) I was thinking am I crazy? Did a dentist actually cause this? And he said I would be feeling better… so the second time I went back, and it seemed like everyone working there wanted to go home. Extremely unprofessional. I was put in a room with what seemed like an extremely old dentist chair and equipment. The dentist then ignored me for 20 minutes and it seemed like a ghost town. It was a different dentist this time who came in and told me everything was fine. It normal for it to hurt.

I felt there was something wrong. I went to a new dentist with amazing reviews and amazing bedside manner. They are extremely professional. They did an x-ray of my teeth and found I filling was put OVER a cavity/decaying tooth. We also found the drilling was SO close to my nerve and my new dentist was astounded they did not numb me at all.
I truly believe this was not right and they just wanted to charge my insurance. I’m now paying out of pocket again, for the 4th time, to fix everything with the new dentist I trust. What should I do to make this right or be compensated for these damages? I am having pain in a tooth again, in one that he drilled. I’m extremely frustrated and would love any input on this.

Additional info is that this all happened in NYC.
Location: New York City