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 or muiltiple of a 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.)
To go into more detail about your exceptional types, it's useful to have at hand both the tables for individual root systems in Bourbaki and the lists of positive roots at the end of Springer's paper here. In simply-laced cases (here type $E_n$), the adjoint representation is quasi-minuscule. This is easy to analyze from the tables, since subtracting arbitrary roots from the highest root usually doesn't give a multiple of a single root. In type $F_4$, the only minusucle highest weight is the highest short root. Here the weights other than 0 are the various short roots, so it's again easy to see that the difference need not be a multiple of a root. In your other minuscule cases (for $E_6$), more computation is needed than I've done.
By the way, since this is a roots-and-weights question, it's mainly about simple Lie algebras (not algebraic groups), whatever the application may be.