From my own experience and having seen a lot of people struggle and also manage to quit drinking, this is the conclusion I've come to:
1. Quit drinking. (By whatever means--I did AA and the steps, but there are other paths, of course.)
2. Deal with the underlying problems.
Maybe it's possible, but I haven't seen it work the other way around. Alcohol creates so many problems of its own, that I can't imagine fixing problems while still drinking.
For me, I also used the idea of me "fixing my problems" as a way to keep drinking. Because there was this underlying sense that if I could fix the underlying "issues," then my drinking problem would go away. Of course, I couldn't fix the problems while drinking, so I kept drinking. It was a vicious loop.
Only by focusing on sobriety first could I even begin to heal and work on the underlying issues.
I wish you the best.
PS I also agree that not everyone who is sober is happy. I'm not always happy. Sometimes I'm flat out miserable. That's life. But I am always happy and grateful that I'm sober, which is different.
If I weren't sober, I'd be dead, so I'll take the ups and downs of life as a blessing.