When we went a few years ago, Dollywood didn't even have "mugs." They had those sports' bottle type that are not insulated at all. We bought some since we went in two different days, but the ice melted really quickly. Maybe bring your mugs to keep your drinks cold--lol
I doubt they take the mugs, but you could ask. When we went, we were kind of an anomaly for Dollywood employees. At the time we went, an All-City pass (Celebration City days) would get you free admission to Dollywood. When we showed our passes at guest relations, the employees kind of used us as a training moment. Throughout the park, employees asked us if we were Gold Passholders. We told them what we did have--All-City Pass. Some gave us the big discount Gold Passholders get, and some gave us a smaller discount.
My point is--be prepared for people to not have a lot of experience with SDC season passholders!