I have never dated anyone, so I cannot relate.
But I have had feelings for single women, and with some I thought there was a chance she would feel the same way. Then I would get really nervous. Not so much meeting her, since usually I already had, or at least had chatted online for a while. No, my anxiety is that my quirks would push her away. I am risk-averse, lazy (other than on the job), do not like to do chores often, and in some things, I am very set in my ways. And simple administrative tasks in life give me anxiety. Like making phone calls to handle personal business, etc.
But if we already knew all that about each other, it would be easier. Like a friendship that morphed into something more.
I am not sure I could just ask out a stranger. I would need to know her first.
Another source of anxiety is intimacy. I have never made love to a woman. I do not think I am attractive. Also, not all my parts work like they used to. I may need drugs. Sigh.
As for dating a woman with anxiety: as long as we could go out to eat, and go to malls and museums, that would be fine. But no concerts, unless we sit far away, due to my tinnitus.
Also, I am an introvert with no social life, other than chatting on here. So she would be my primary social life! Of course that may be too much of a burden for her!