The adage "It's in women's nature to flee if you chase them, and to chase
you if you flee" is one I've heard, read about, and seen in my own
experiences numerous times.

And I think I have a pretty good grasp of this idea and understand it - I'm cocky/funny, am indifferent where appropriate, and try to be unpredictable (spend tons of time with her one day and then have minimal contact for the rest of the week), etc. But I need some help:

My issue is that even though I feel I've established a great rapport with
the girl I'm interested in, when I start to show my affection, she starts
to move away from me - she becomes more sassy, doesn't contact me as often as she normally does, and in general I believe that at these times, she has more 'power' over me and she's aware of it. She becomes the 'prize' in the relationship - I lose my initiative.

The thing is, this happens on and off - I'll have control of the 'flow' of
our interaction, then she will, then me again, etc. This has been going on
for about a year! I want her to be my girlfriend, and I think I already
have a great rapport with her, but I feel that if I take the first step to
make the relationship more serious/formal, she'll just metaphorically move further away from me.

How do I close this initial stage of interaction with a woman, and progress towards a relationship? If the saying above is true, how do I become intimate with a girl at all? If I show or tell her that I'm interested in her, won't she just back off?

Sorry for the long post - I feel like I'm stuck in a conundrum. This
problem's been bugging me quite a bit; it's come to the point (for a while
now) where though I want to message her occasionally, I just don't,
simply because I'm afraid that by messaging her and initiating contact
first, it'll be seen as a sign of neediness or dependency and I'll push her
away. (Somewhat painful for me, because I don't see her messages in that light..)

I was wondering whether y'all had any advice or could help me out regarding this situation...

Warm regards,
a random dude