By entering a plea of not guilty at your Arraignment, you will automatically trigger a subsequent court date. It is at this stage of your prosecution that you can request the appointment of the Public Defender if you are without financial resources to hire a private attorney.
Have your attorney request Drug Court as part of the Plea negotiations
After the Araignment date, your attorney should be in a position to determine whether Drug Court is appropriate. Assuming it is, your attorney should petition the State Attorney for your case to be pre-screened to determine whether the specific charge is qualified as well as whether you otherwise qualify, ie. live in the courts jurisdiction, have no prior felony record. Once you have completed drug court, your case should be dismissed.