Advice needed, and appreciated

I have been in a relationship with a woman who is divorced and was cheated on, as was I in my marriage (I am also divorced), for a few years now.  We have had many bumps because of not only our marriage issues which resulted in trust issues, but due to both of our childhoods (she grew up in a crackhouse but now owns her own business, I was treated as if getting good grades and doing what I was supposed to was a sin because my step-brother was always in trouble and my mother acted as if I was trying to show him up, which as a child was never even a thought in my mind), we have had to work through many problems.  One of my issues was because as a child I would get in trouble for doing good things, I basically began hiding good grades, and anytime I did make a mistake I was treated as if I was the worst kid in the world so I also had problems with not being honest if I DID make mistakes. A wonderful therapist figured this out about 2 months ago and I am on the road to a much happier life now.  My girlfriend (I'm 53 she's 40, both very active and in good shape, like to enjoy life, ect...) and I got into a big argument Friday night because I apparently knocked her phone off of a table we were sitting at in a music club and instinctively picked it up and put it back on the table.  I noticed her screen protector was broken, and she said they break so easy.  About 15 or so minutes later, it seems a friend of ours who was there told my girlfriend I dropped the phone, which obviously I did but i sincerely don't remember doing it because we were indulging in adult beverages and just having a good time listening to the band.  When my girlfriend asked me if I dropped her phone, I told her honestly I guess I knocked it off the table but I really don't remember doing it, and she freaked out, decided this was me once again lying about silly stuff, and that night ended horribly. 

  Now, I am torn. I love her in a way I have loved no other (I know, it sounds corny), and she and I could be amazing together. The issue is I have been a certain way for so long that I understand she had doubts about me finally learning how to fix a problem I know now that I have (thanks to a great therapist), and I can't "prove" to her I"m not that person anymore because she seems to not want to take the chance and believe that.

  SHe is not talking to me, and as I said, friday night was a bad bad night.  We both said things we shouldn't have, her out of anger and hurt me out of frustration and hurt (not namecalling or anything like that), so now I have two choices. I can let her go, and I honestly was going to try and do that but yesterday I had a long talk with a mutual friend and she said she knows me too well and that in my heart I don't want to (she's correct).  Or, I can not give up, and just let her know I love her dearly and I really wish she'd stop and try and accept I'm not doing the stupid things I used to.  She has a bad habit of just giving up on things because of where she came from and being let down not only as a child but as a wife to a husband to whom she gave her all and he cheated on her, verbally abused her, ect.. and I know that's what this stems from. 

  I won't lie, I wish I could hear some positive reinforcement, but I do value everyones opinion here. I love this woman like no other, and we could be so happy if we just worked together and not against each other. We truly are at a crossroads in our relatioship, and despite doing things that caused both of us to feel some resentment to the other, I feel we need to remind ourselves of where we came from and, in spite of that, work together to keep moving forward and be as happy as I truly believe we can be.  Thanks to all in advance for your thoughts.