You are very brave for both acknowledging and challenging yourself. I am very glad that you have taken the opportunity to ask this question.
The very simple answer is that you do have a drinking problem. In many ways it is classic in that your thinking revolves around it, you use it as a release from the anxiety, others have noticed the volume, and it disturbs you that those who should take an interest haven’t. The truth is that the training for most therapists is extremely poor when it comes to an accurate assessment of alcohol use and abuse. Very few have had any specialized training and their experience is limited to what they may or may not have had in their internships and classes. It is not a required part of the training for licensure as a therapist. Those who have had that training or certification have typically pursued it as post-graduate education.
This doesn’t mean that the therapist you are seeing isn’t competent or lax. What it does indicate is that his or her strength may not be in substance abuse, just like you might go to a general practitioner of medicine who may come across something that isn’t part of his or her expertise.
In this case the solution is simple. I would encourage you to attend at least six Alcoholics Anonymous meetings in your area. If your therapist had responded to your drinking issue this is likely to have been the recommendation, and it is the recommendation that others who have struggled with it would give you. The above link will direct you to a list of meetings. Find the open ones and go the six times before making a decision.
Let your therapist know that the issue has evolved to the point where you have decided to get some support through AA. Further I would tell your therapist about your disappointment in her not seeing your need. My guess is this is where the real work in your therapy is, and your therapist will likely pick up on that and help you through it.
Finally I would not simply move on to a new therapist. This is a real opportunity for you to work through an important dynamic about not being seen adequately, and leaving for another therapist won’t allow you to seize this opportunity.