I want to post an update. Basically, I completely gave up on trying to play with her. I just worked on standing and sitting by her and the only time I ever tried to pick her up was if I had to put her back in the cage, so I always used millet. Today, she is way better with me than she was when I posted this question. She now recognizes my hand as a 'tool' to get to places she wants to (a very good thing). Whenever I open the cage she'll always hop onto my hand to get out too. I've been working on getting her to stop biting my hands whenever she sits on them and it's working. She still does lightly sometimes but nothing like at first (she would literally make me bleed for no good reason. It wasn't an aggressive bite but just of 'i want to chew on your hand cuz I feel like it' type of bite). She still doesn't like to 'hang out' with me by sitting on my hands or shoulder, but she does enjoy climbing on me sometimes to chew on my clothes or hair. Sometimes she even flies to me now when I don't expect it, but she still doesn't stay for long. But it's nice to see that she actually wants to be by me now. There are moments when I'm leaving the room and she starts to chirp and fly around just like my previous boy did when he didn't want me to leave.
So overall, things are getting better. I guess I just had to back off and let her choose what she wanted to do. I think she'll get more social in the future too.
Since she isn't as social with me as my other budgies, I'll be getting another budgie for her. I didn't want to at first, and I know some budgies are fine with being alone, but I can tell she will be happier with another bird. I might even have the opportunity to handfeed a budgie (I'm not sure if I will though) but I'm hoping the next budgie I have will be a very social one. With my previous two, they helped each other become social so I'm hoping the same thing will happen here.