A mutual friend of my best friend of six years introduced herself to me. She initially showed a lot of signs that she was interested in me. She was always the one to initiate conversations with me, complimenting me, etc. She kept introducing me to only her female friends. This went on for several weeks.

However, once I asked her to prom, things started to get out of hand. She first thought that it was "really sweet", gave me a hug, etc. But she came up with a convenient excuse for not being there, which was "going out of town". I still can't confirm if this is a lie or not, but if it was a lie, that's a pretty big one to tell (apparently she told all her friends that she's going out of town too). I asked her to hangout sometime, she said "yes whenever I have free time". Even after this, the whole "initiate conversations and complimenting" continued until a few days after me asking her to prom she turned cold and stopped doing her usual routine (well, she didn't stop it completely, but it was much less than usual and she gave me a cold shoulder for the majority of the day). I then asked her to hangout again online (stupid mistake here), to which she replied that she was "busy all weekend". Using a psychology trick I heard about online, I asked when she was available, to which she replied "I don't know, but I'll tell you when".

What in God's name is going on?

Maybe you mis interpreted what she was doing. Maybe she was just being friendly. Maybe she treats everyone like this.
She obviously is not liking you that way so just back off