The way to have made things right was to have waited for a case to have been filed against you and then to have hired a lawyer and let him try to negotiate a deal or fight the case for you. You can still do that, but you've hurt yourself and it now won't be as easy.
When you are the suspect in a criminal
offense you should never talk to the police when they want information. If they have enough for an arrest, they're going to charge you anyway, but the last thing in the world you want to do is help them out, especially as they may not have had enough information to arrest you.
Now they do. And you have not only told the police that you were involved in this incident but told them that you were driving on medication that impairs your judgment, which is potentially a driving under the influence charge too.
This charge is not a traffic violation. A hit and run offense is a criminal charge. If there were no serious injuries or serious property damage it's a misdemeanor
but can be a felony. Additionally, a conviction, besides giving you a criminal record
, will cause a suspension of your license.
So do not call the police again. If they call you, do not discuss what you did or didn't do. To do so is to make a confession and tie your lawyer's hands when he tries to get you out of this mess. If the police want you to come in to be printed and booked, do not go down there to see them unless/until you have contacted a local lawyer who will instruct you as to what you do and don't have to do.