Have you ever considered starting these conversations on the internet? That would allow you to get to know someone without the same sense of urgency of reply (if you private message or email, rather than chatroom or whatever) so you can think of something to say before you do it and have a chance to think before you send... and if you get too anxious you can politely say you have to go "dinner is ready" "there's someone at the door" or whatever and she'll be none the wiser.
You might also like to consider researching any local self-confidence courses or related, which might be useful to you in at least bluffing how anxious you are feeling.