We bring sanitizing wipes, and clean out the drawers, and then dry them quickly with a clean hand towel from the bathroom.
We then line them with tissue paper (saved and recycled from shopping/gifting) or wrapping paper (either recycled from big gifts at the holidays, or bought on clearance after the holidays for this express purpose) or if I forget to pack drawer liner of some form, then we just ask housekeeping for additional clean towels to line the drawers.
All of this takes (easily) less than 10 minutes. Then we unpack into the freshly cleaned/lined drawers; we have done the "ziplock bag" packing method with one complete outfit per bag/per day for *years*, so that's really quick. (we recycle those bags, BTW - I have some that I'm packing now for our next trip that are easily 5 years old LOL)
We keep a kitchen garbage bag (with drawstring) in the closet for dirty clothes; we line a "pop up" hamper we use just for travel with it. It's re-used as the dirty clothes bag for the trip home (it's already dirty!) and then the hamper stays clean to bring our clean, folded clothes back to the room in. We do laundry once during the trip so we only have to bring half as many clothes (SHOCKING NEWS: people *might* see us wear the same clothes twice at Disney World! Alert the media!) and when we get home, it all gets laundered again because it's all dirty, and because we brought just enough clothes... for the second half of the trip after we do laundry...! Laundry doesn't take long - we typically break it up into several smaller loads, including the dryers, and the whole process is 2 hours or less out of our trip, but worth it for us because we can pack less. Plus, we typically have at least one Resort day built in to every trip, anyway, so it's no big deal; we just make sure to plan that for the halfway point.
Been going to Disney World since 1972. Never have seen a bedbug at any Disney Resort hotel, never have had any kind of bug or critter problem at any Disney Resort hotel (unless you count our "frog friends" at WL who stick to the glass windows in the autumn or all of the little skinks at POFQ that love to dance around your feet LOL)