How long do they have before my case in dropped?
Driving under the influence of alcohol, drugs or a combination of both is punishable as a misdemeanor which has a one (1) year statute of limitations.
Incidentally, it's not a question of your "case being dropped". No case has been filed to be "dropped".
In the event, the DA's Office in your county files a criminal complaint charging you with a DUI after the expiration of one (1) year from the date of when the offense occurred, you don't just walk into a court and expect that the judge will automatically dismiss the complaint. A formal, written motion is required (assuming the DA will not voluntarily dismiss the case) and you will need an attorney to prepare the motion on your behalf.
You can always contact the county district attorney’s office to determine what happened to the charges. If there has been a decision not to prosecute, they will inform you of that. It is probably safer to go online to the court’s website and check the status of the citation in that way.