Welcome to the board
and big HUGS to you. You are dealing with a lot of difficult things right now. Alcoholism is a hard enough thing to deal with. Having sexual addiction added to that just isn't fair.
You are the only one who can answer the question "Why do I stay?" Sometimes it's hard to answer that one honestly. But you might want to give it some thought. It's hard to live in a relationship that brings you hurt feelings and frustration on a regular basis.
You deserve a happy life, we all do. One day at a time, you will get there. Maybe you can start by thinking about what a happy life would be for you. And how you can work towards getting there. Be gentle with yourself, this is not always an easy task. When you are feeling overwhelmed by what you are dealing with, stop and say "I deserve a happy life. What small thing can I do today to work towards that goal?"
In the meantime, stick around. This board is full of wonderful people who offer support, encouragement and lots of HUGS when you need them the most.
Peace,
Gabe