I think you are being way too hard on yourself. Perhaps you have this feeling that if you beat yourself up enough, she’ll see that, and tell you to stop it. She’ll forgive you as a way of getting you to stop hurting yourself.
Frankly, I don’t think what you did should have caused this much trouble. I would hardly call it cheating. She was an ex. So what? Don’t you get to talk to anyone about your relationship? Or is your current S/O going to keep you isolated from others? If this is the way she is, then I think this could be a blessing in disguise.
She is too possessive and clingy and insecure. You might possibly have a codependent relationship that will repeat this pattern over and over unless you guys can find another way to relate to each other than through secrecy and getting caught.
You may have left your email open accidentally on purpose. That is, in one part of your mind, you desperately wanted to talk to your S/O about what is bothering you, but you didn’t know how to bring it up. So this part of your mind overruled your conscious mind and made you “forget” to close out of email, hoping she would read it and then you could start to talk about it.
But then, the conscious part of your mind still wants to evade the issues, and so you beat yourself up for the “inappropriateness” of what you did, which sidetracks your discussion from the actual issues.
It really doesn’t matter who you spoke to or wrote to about it. What really matters and what you should be apologizing for is that you didn’t have the guts to talk to her about what was really bothering you. Even then, you don’t need to apologize. This stuff is hard. She should want to know what you are thinking. You should be telling her what you want and she should be talking about what she can give, and if you can’t get what you need from each other, then you should be talking about splitting.
You didn’t cheat. You just didn’t know how to talk about what you need to talk about. Your current self-flaggelation is not helping. It only sidetracks you from your real issues. Talk about your real issues. Set aside the bullshit about cheating. This is not a pattern you want to set up in your relationships because it will happen over and over again, and you’ll never get to talk about what you need to talk about and you’ll always be semi-miserable because of it.