Hi Natalie, welcome to SR! WE can't dispense medical advice but I can share my thoughts based on experience.
You are shaming yourself; stop denying yourself of the resources available to help you treat the disease of alcoholism. It's no different than denying yourself the resources that have treated your cancer.
Tell your Psychotherapist that the root of your mental issues may be alcoholic in nature, they can get you the appropriate treatment. Your employer can not by law retaliate against you; and they should have an EAP(employee assistance program)to help you find treatment while you maintain your anonymity in the workplace; anything short of that and your employer becomes available for a lawsuit.
You CAN do this, you've alredy proven that and more about yourself through your life journey.