Glad you are here and welcome to SR!
I worked in the health care field for the first 6 years of my sobriety before I retired from it. I originally had the same concerns as you have but what I have learned is that most people feel the same way I do and want their anonymity protected. So for them to say they saw you in a meeting they would have to explain why they were at a meeting. This significantly decreases the chances anyone will say anything about you being there. I have not had any problem over the years with it.
Hang in there and don't let your fear keep you away from something that might be a lifesaver for you. I know without the program of AA I would probably not be alive as alcoholism was killing me. After 20 years of drinking there was not much left that I had not tried to stop drinking except AA. It was the end of the road for me. Either way I do hope that you are able to find the sobriety you seek.