I've been a full timer for almost 10 yrs and have "camped" my entire life. While campgrounds don't allow washing dishes in their bathrooms, I seriously doubt they'd have a problem with you washing a cup.
As far as RVs go I've owned everything from a truck camper, travel trailer, 5th wheel, and both class C and A motor homes in my lifetime. Other than car camping in an unconverted truck or van I've never, ever seen an RV of any kind that didn't have a sink.
As far as you're own unit - How do you wash your hands after going to the bathroom? Even Class B\CamperVans have small sinks. What are you camping in?