I've been where you are, And I know exactly what you mean.
But let me tell you some truth that nobody seems to mention.
Having anxiety or not having it doesn't make you attractive or unattractive.
Girls look at other things, and if they are attracted to you, your anxiety doesn't make you less attractive. not even one bit. In fact it makes it look cute.
Tell her "I don't do this regularly, this is kinda weird, but you look so nice that I had to talk to you
"
She will sense your anxiety but she will be more encouraging, she knows you are authentic. Not a player without a heart.
Be thankful for your anxiety.