You probably know that I don't tell people what choices to make. Instead I help people resolve the inner blocks that prevent them from making their own decisions.
When you ask me if you should try to work it out, I know that you think that the real issue here is to stay or go. But, I've got to tell you that I think the issue here is about why you would even consider sticking with an unfaithful alchoholic who shows no signs of wanting to improve himself.
Many women stay, even when their relationships are sinking ships. You need to find out where you learned that you are supposed to stick with a scene that is bad and unhealthy for you.
Most of us had mothers who were martyrs. All of us have watched our mothers and grandmothers give up their lives to help, nurture and support other people, even when they suffered for doing so. Not a healthy thing to do.
Figure out where you got the idea that you should stick around when your needs are being violated. Then, do an attitude adjustment, and remind yourself that you are supposed to take care of number one and stay in your comfort zone.
Good luck. I hope you decide to take care of you.