My girlfriend went through my phone while I was sleeping, and saw texts I sent to a mutual friend of ours about another girl, whom all three of us know. The texts said how I found her cute and physically attractive and that her demeanor and voice are also attractive.
Obviously when my girlfriend read these it was unsettling and was very wrong on my end. She wants to work on our relationship but it’s hard. She wants to know if I am willing to change and make it work. But I am stuck between breaking up with her because she deserves better than me, or working on things.
LemonVibe 
Break up with her. First off you didn’t do anything more than notice another woman was attractive and I don’t consider that cheating. Secondly she breached your confidence and privacy by going through your phone while you were asleep. But there’s more to that isn’t there or you wouldn’t be thinking about breaking up with her for this at all. She doesn’t trust you and you can’t trust her not to go through your private messages.
Something is making her feel inadequate and it’s most likely you both are not a good match. Her gut is telling her so. Yours is too.
I wish you the best of luck. I see drama in your future.