One month and already you think you two are bf-gf?
This shouldn't happen until you two have been dating 3-6 months.
Also, the first YEAR of dating is the trial period, during which time anyone can lose interest in the other.\
Sounds like she is losing interest.
Yes, actions do not lie. Words can lie, but not actions.
Give up on her? You don't even HAVE her yet. That takes about a year of dating before you can assume that the relationship MIGHT last.
Of course you know what to do. It is what you don't WANT to do. Which is to face reality, let go and move on.
And people rarely change. No matter how much talking goes on. Because most of our behaviors arise out of subconscious behaviors, and it takes a long time to reprogram our behaviors .. years even. Assuming we want to and know how to.