Yes, I'm afraid they will show up, unless you obtain an expunction. That may or may not be able to be done at this point. Within the first two years, expunction of a misdemeanor under those circumstances will be within the prosecutor's discretion whether to agree to or not. Once the two year statute of limitations is up, you will definitely be able to get the case expunged, but even then it's not an automatic thing. You'll need to have a lawyer file a separate civil case requesting the records be expunged, and also inquire about the possibility of doing it early, if you'd like to try to do that. Good luck.
It will remain public until it is sealed or expunged. If this was truly an outright dismissal, you can expunge it completely two years from the date of dismissal. If the dismissal was the result of the successful completion of deferred probation, you are not eligible for an expunction but you are immediately eligible to seal the record. Any defense attorney in your area can do this for you.
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If you did 24 hours of community service, sounds like you got a deferred adjudication probation as opposed to an outright dismissal. If this is correct, you can petition the court for nondisclosure which will hide the record (if granted) from most people (although not licensing agencies or law enforcement.)