You should talk to the person you feel most comfortable with, and that seems to be your advisor. Maybe it would help if you wrote down what had happened at the Disability Center and what you would like your advisor to help you do. Clearly anxiety is another one of your "disabilities' so it okay to ask for help with that too!
So right now, what is your goal? to let someone at the disability center know about the issues so they can be addressed? Or just to have someone you trust help you go to the disability center? Or for someone to talk to someone over there for you? To have someone arrange for note taking for you?
Even if you don't know, you could talk about that with your advisor.
I would suggest saying that you know you need to avail yourself of the disability services, but have had some poor experiences there. (explain them). Then say that you are anxious about going, so what could they suggest?