Ano,
I have had absolutely no problem going to the bathroom for years without touching myself. And, with No accidents. When you lower your zipper all of the way, you can then kind of pull your uderpants to the side, or even open them up where they are meant to be opened, and then just kind of lean forward,and this should allow everything to move automatically. You just lean forward over the toilet. Or, if you are at a public urinal, then you can also use your underpants to support yourself from underneath, without touching yourself at all. It's kind of like using your underpants like a sling from underneath to do all of the supporting and re-positioning.
Also when showering, you need not touch yourself at all. If you just put lots of soap up above, it will all slide downwards to where it needs to be together with the hot water.
I in fact once asked a talmud chachom from Toldos Aron (who are very machmir about tevillah in a mikveh). What does one do when they really are unclean down below. (either from let's say a wet dream, or any other reason), and his answer was that we do NOT touch ourselves down there, even with the pupose of cleaning. He said that when one immerses in the hot mikveh, everything will come off by itself. (The reason I had asked was that I believed that the dirt would cause a chatzizah in the mikvah)