I heard the other day that the number one symptom of alcoholism is depression, drunk or sober. Probably closely followed by anxiety. It is normal for alcoholics, especially in the first we while of sobriety. I couldn't have been more miserable, depressed and fearful when I sobered up.
However, I got into a program of recovery and it did not take too long for those symptoms to leave and be replaced by a sense of purpose and a feeling I was on the right track.
A small percentage of alcoholics have coexisting conditions which may continue to manifest after the alcohol has gone from the picture, and progress he's beeen made in recovery. With the alcohol gone it is much easier for the professionals to treat other disorders because the symptoms can be more easily seen, and of course the subject is more likely to follow directions with a clear head.
So yes it is normal and the problem may vanish if your alcoholism is treated. If it doesn't, it is wise to consult a professional.