What can happen to my insurance rate after the DMV hearing but before the court hearing?

During a DWI checkpoint, I had to do the FST and then was taken to the station for a breathalyzer. The machine did not register so I was arrested for "refusal to blow". My DMV administrative hearing was last month and had gotten recalled because the cop did not send in all of the correct paperwork. As a result, my license is not suspended and I do not need to get SR22 insurance. My next court date is next month, and I am hoping to have the DUI charges reduced. In the meantime, can my insurance rate go up since I have not been convicted/charged for DUI? And if my charges do get reduced, will the DUI still show up for insurance companies and will my insurance rate still go up? Thanks.