one step forward 1000 steps back.

I cheated on my girlfriend, and she found out on my personal online journey.
Now we agreed to work on this relationship, but every single day her emotions are up and down.
She always says, “Why did you cheat?”
“I think you would do it again.”
“I just don’t love you anymore.”
“I love you.” “I want to make it work” “ughh HATE YOU!” “You’re only with me because you can’t be with me anyone else.”
I realize I put myself into this position. She’s in North Dakota, and I live in NY. The next time we’re starting to understand each other is June. This is baffling for the both of us because I can’t be with her physically. Altogether I can say, sorry, all I can make out is try to answer her questions the best way to honestly.
I am past the cheating. I am sorry. I want to so badly to make this work more than anything else. I desire to be part of her life, and for her to be part of my lifetime. I am a female, and this is my first time ever even dating an adult female. I was mixed up, and I hated the first time me and she had sex, but after bringing more into the sex. I terminated the affair with the ex boyfriend, and I haven’t really even thought of cheating.
It’s forcing me off the ramparts. A piece of me wants to tell her, “LET ME GO THEN!!” while the other part of me like, “I deserve this.. If I desire to get this work… I need to hear this. I ask to aid her through this pain I’ve caused.”
What should I do different? :/ Would letting her work make her happier? She desires me to move to N.D with her. Merely, with school, my occupation, my car, and my dog. I simply can’t move and simply let her support me. (she’s in the air force and lives on base). I stated yes, but if everyday her emotions are changing, I want to stick around here in my home town and not be under her thumb.