first off, the clean handle should be on top. the nasty handle shouldn't drip onto the clean handle.
second, as you are leaving a public bathroom, reach under your shirt/jacket (hopefully something untucked. ) use the fabric as a barrier for your hand and grab the handle with the front of the shirt/jacket.
If you are lucky enough to have a paper towel dispenser, dry your hands with one, use it to open the door, then prop it open with a foot while you throw away the paper towel
Ok. Your first one is “tomatoes, tomatos.” The second is my go-to solution of sorts as well: use a paper towel, open the door, and dump the paper towel in the nearest bin. This works more often than not.