I don't think anyone in this thread has laid it out explicitly why referencing a DUI is a terrible idea, so I'll stress it here. You were incredibly lucky that your DUI occurred when you were a minor and that you had sufficient legal representation to limit the damage it would have on your record. If you were 18+. received the same charges, and it got on your record, you would have to spend years distancing yourself from the conviction before application. Even after that, your application would likely have been dead on arrival to a good proportion of schools (why should they take a chance on you when they can admit 50 other applicants without a DUI on record?). The fact that you are able to apply without that potential DUI is incredibly fortuitous; you should never bring it up in any part of your primary, secondaries, or interviews. And just one more time for emphasis: A DUI is a charge that would significantly harm any application regardless of how amazing the rest of the application is. You are lucky that yours occurred in a context which would allow you to not report it, and you should never offer this information up voluntarily to an admissions committee.
Secondly, while you definitely may have learned a lot from the experience, the reasons you have offered here to include in a potential adversity prompt would reek of entitlement when read by an AdCom member. A few examples of the essays that would come in for adversity prompts include food insecurity, homelessness, discrimination, and other such problems. If you write about not having a car or relying on your parents to help you with basic transportation needs, it would come off at best as a privileged applicant who is out of touch with problems that their future patients would face and be viewed worse if the AdComs' imaginations start running wild (DUI, reckless driving habits, overdependence on parents, etc.).
Unfortunately, this is one of those cases where something that may be important for your growth and development does not function well in a med school application. You need to go into your application with the mindset that you should never reference this incident and think of alternative topics to bring up for your essays.