Your age is irrelevant. The biggest thing that concerns me in your narrative is the "dealing with alcoholism then" part. Unfortunately, many people don't "deal with alcoholism" once and then never have to deal with it again. And being an impaired physician is a serious risk that could not only put patients in danger (which is what any state licensing board or hospital will care about), but could also endanger yourself (which is what I, as a person who treats addicts, would care about).
First: have you ever had any kind of legal trouble because of your drinking? If so, you should start by contacting your state medical board and finding out whether you would even be eligible to be licensed as a physician some day. If the answer is no, it's best to know that now, before you sink any further time or money into pursuing a medical career. If the answer is yes, be aware that you may still have to register with your state board or otherwise do additional paperwork, drug testing, etc to prove your sobriety. Every state has its own rules.
Second: assuming you are able to be licensed, what kind of plan do have in place to protect yourself and avoid relapse? You have obviously learned some improved coping strategies during your law career, and people will definitely want to know about what you do now to cope with stress and obstacles that you could continue to do instead of using alcohol?
Third: do not, under any circumstances, lie to adcoms about your past with alcohol problems, especially if there is a record of any kind. But at the same time, be aware that it will be a major red flag, and yes, you will have to be a significantly better applicant than most everyone else in order to hopefully convince someone to give you a chance. It's ironic, but there is a significant bias against recovering addicts in medicine. You should know that fact up front.
Finally, it would help if we had some understanding of why you want to go to medical school, and what your goals are. If you say you want to be an anesthesiologist, I will tell you now that this is a very dangerous path for you (or for anyone else with an addiction problem) to walk, since anesthesiologists have the most direct access to controlled substances of any specialty (and the highest risk of dying from addiction/drug use). But as any type of physician, you will still have the ability to prescribe substances that are much more addicting and dangerous than alcohol. It is worth seriously introspecting up front about whether this is a risk that could leave you dead.