I know your ex broke up with you because she didn't have enough time for you (which, in that case, I wouldn't have wanted to stay in contact with her any further) but did she question you a lot about other girls while y'all were still dating? It seems like maybe she was concerned about whether you had been with other girls and you denied it. That's the only reason I can see why should would be upset and call you a liar... because she thought the 2nd girl was someone you had probably been creeping with during the relationship. You should probably clarify that with her (if you still care).
As for the 2nd girl, you are probably just trying to see her too much. She might know that you are seeing her as a potential S.O., and she doesn't really want that type of relationship with you which is why she's trying to keep the distance. You can always take the straight-forward, direct approach and ask her what's the deal with you two.