The State has 2 ways of bringing misdemeanor charges: (1) by citation, and (2) direct complaint with a summons. For non-traffic related offenses (such as possession of drug paraphernalia), it is common that the State files charges by means of a direct complaint with service of a summons. For misdemeanor offenses, they have 1 year from the date of the offense to serve the summons.
In your case, as you described it, they have served you within one year, and they have brought charges in compliance with Arizona law. You are NOT likely to get this case "dropped" because "there is no signed documentation for this incident," if you are referring to not having a signed citation - the State doesn't have to have one if they have properly served a direct complaint via summons.