Hi Andy, welcome
Well opinions are subjective so I'll give you mine. You are likely to lose this girl.
Her parents are being parents. They like to know what kind of guy their daughter is dating. Simple, maybe a little early but they are being cautious and caring for the child they brought up. They are not causing you to be more committed. Neither is your girl.
You hit it off really well and you don't want to be alone all your life. So isn't it reasonable for someone that likes you to take that step of introducing you to her parents? I think it is. Pity many other parents don't give that kind of care.
I'd visit her home, break the ice with her parents and after a few more dates if things are still going really well start to set a small plan. This should be about the time when you are falling in love. Remind her that you want to date for a good amount of time before commitment. If she asks how long that is then you could hint a timeframe "maybe a year or so" for example.
I've had friends in the past that were in love and all was fine but their timeframes were out of synch with their girlfriends timeframe and they ended.
Your anxiety will drop over time the more you take her out. But, in my opinion, severe anxiety (which is from your own mind) could cause harm to your relationship. Go with the flow and enjoy your time with her. When you feel comfortable be honest and tell her you are nervous/anxious. Get past it together.
I haven't mentioned her "disability" because it is totally irrelevant whatever it is. And tell her so.