You only feel normal when you drink because you are addicted to alcohol. You will have to stop drinking to see what else is wrong with you. I would suggest going to rehab where they can safely carry you through detox and help you find a safer alternative than alcohol to deal with your emotional problems.
I found that anxiety is at the root of a lot of my addictive behaviors. I was diagnosed with Generalized Anxiety Disorder recently and put on anti-anxiety medication. It made a huge difference. I have always been very neurotic and anxious, so I thought it was normal. I have also always been addicted to something (even as far back as 3 years old).
Alcohol for me thought is all about treating depression. I crave alcohol constantly when I get severely depressed and without being on anti-depressants, I have a feeling I would be fighting cravings for alcohol regularly. I didn't mean to get addicted to alcohol or to use it to treat my depression, but I found out it worked when I got put on probation and had to stop smoking weed. Now alcohol it my go-to drug.
I don't know how much you can relate to my story, but not feeling normal unless you are drinking is a classic symptom of alcoholism. Nobody feels that way unless they are an alcoholic. Is the drinking causing you problems or is it working for you to fix your anxiety? If it is causing problems, take my advice. Have your therapist help you find a good rehab, try getting sober and staying sober for a while, and then try some other means of dealing with your anxiety.
My anti-anxiety medication has way fewer side effects than the stuff I was using to treat my anxiety before (marijuana, alcohol, nicotine, caffeine, sugar, fattening food, compulsive spending, etc.). I'm sure you'll have similar results if you get sober.