Can a diet drink like Diet Sprite Zero with alcohol sugar test positive for alcohol after being opened then sealed for 7 - 10 days?

Short answer, no!
The term alcohol sugar or sugar alcohol is a marketing gimmick to make something chemical sounding sound a little more consumer-friendly. The name is derived from the fact that part of their chemical structure resembles sugar and part resembles alcohol. There is, however, nothing in common between the alcohol in Alcoholic Beverages, which is ethanol and sugar alcohols which are typically sorbitol, lactitol, mannitol, xylitol, isomalt, maltitol and hydrogenated starch hydrolysates (HSH).