EDIT: Reading the question more carefully, I think the difference between the highest weight and an arbitrary Weyl group conjugate will almost never be a single root.   (What's true is that the difference between "adjacent" weights in that orbit across a single reflecting wall will be 0 or else a root.)   The adjoint representation in type $E_8$ illustrates this behavior.   The saturation property in Bourbaki implies here that weight strings between such adjacent weights are of length 0 or 1.