You were not convicted? You can honestly answer "No" to the question on the 4473 about whether you were ever convicted of a felony. The fact that it shows up with no info on a background check might mean a delay, but it shouldn't result in a denial.
The larger question, I guess, would be how you plan to answer the question "Are you an unlawful user of, or addicted to, marijuana, or any depressant, stimulant, or narcotic drug, or any other controlled substance?" But that's on the 4473 for buying a firearm, not on the carry permit application. Submit your application, and if you get a question back just tell them the charges were dismissed before the case even went to trial.