Many, if not most people are anxious about beginning, having, and ending relationships - whether it is casual dating or more serious commitments. What we all try to do is manage the feelings as we plunge forward.
One suggestion is to talk to your new friend about your feelings - she is likely a bit nervous too. It might be a good way to build trust and intimacy.
I'm unsure if the therapy you tried in the past was cognitive-behavioral. That is one of the best ways to learn about, and handle anxiety - what you are describing (the worry, dread, pacing etc) can be better managed with some training in this area.
Let me give you an example: After a date, let's say you haven't heard from her in a few days. Make a list of all the potential reasons she hasn't called or texted you - the good and the bad. It might be hard for you to come up with good, or realistic reasons.
People wilth anxiety have a habit of quickly seeing the bad reasons, and have a harder time finding positive explanations for things. Such as, "she hasn't phoned me because she is traditional, and thinks the man should set all the dates".
This type of exercize will help you challenge yourself to see things differently. Yes, it will feel strange at first. But if you come to realize there are a number of potential reasons for other people's actions, you begin to lessen the hold you have on negative interpretations. Try it, and see if you are able to come up with a few examples of potential reasons for people's actions.
If this sounds like something that makes sense to you, you might want to look in your local library for books on cognitive behavioral therapy - such as the books by David Burns. I also recommend Mind over Mood, by Greengerger and Padesky, or the Anxiety and Phobia Workbook by Bourne.
If you like the approach, you can search for a CBT therapist in your area, if you would like more coaching.
Best of luck. Thanks for writing, and I hope I have been helpful.
Mary Lynn Trotter MSW RSW