How do I deal with sexually assaulting someone?

First of all, I'm not looking for any validation or for anyone to downplay the situation, I want to face what I did but I don't know how to go forward.

What I did: Me and a girl have been friends for a few years now. I've had feelings for her, we have flirted tons of times and have come close to dating. One time, I went out with her and a group of friends to a club, and after drinking a bit (THIS IS NOT AN EXCUSE FOR WHAT I DID, JUST PROVIDING CONTEXT), I tried to get with her. I approached her and we started talking, I put my arm around her and eventually a hand on her leg. She didn't SEEM (emphasis on SEEM, from my twisted perspective) to be bothered by this, and eventually went away to do other things. I really wanted to be with her, so I tried to approach her again throughout the night, just trying to talk to her, but she always walked away whenever I tried to do so, I was an idiot and didn't understand why. The next day, I asked her why she was avoiding me, and she told me that she was uncomfortable with me touching her that way, and with me approaching her multiple times despite her clearly wanting to distance herself from me. I apologised profoundly, and have felt awful about it. I've never touched a woman since and haven't gotten drunk, but the feeling doesn't go away.

She still talks to me and has asked me to hang out with her, but I feel very uncomfortable with it.

I don't know what to do. I feel like I have to take responsibility for what I did, but I don't know what else I can do other than apologise, reflect on my actions and never do it again. Victimising myself and labeling myself as a monstrous sexual abuser won't fix anything. It's been months and the feeling doesn't go away. I feel disgusting and I don't know how to deal with this. Sorry if it looks like I'm running away from what I did and trying to find excuses, or trying to get sympathy by acting like a victim. I'm genuinely looking for help and I want people to be honest with me.

Edit: Sorry for using such a strong term as "sexual assault". I thought that what I did should be classified under that term, but, looking at the comments, I probably exaggerated due to inexperience. I did not mean to generalise sexual assault and take away from the actual meaning of the term and the experience of victims who've suffered from it.