There is a finer approach to the dimension of the quotient when the action of $G$ on $M$ doesn't define a fibration, but has singular orbits. You'll find it here. Of course for the principal orbit you have your formula, but on singular orbits it gives something else. I didn't give a general formula (I should do it however) but I treated the examples $\Delta_n = {\bf R}^n/{\rm SO}(n)$. Topologically, for any $n$ these quotients are homeomorphic to the half line $[0,\infty[$, but not diffeologically, and if we have well (according to your formula) ${\rm dim}_x(\Delta_n) = 1$ for $x \neq 0$, we have however ${\rm dim}_0(\Delta_n) = n$; which proves by the way, that for 2 different quotients, if they are homeomorphic, they are not diffeomorphic. I'm not sure it will help you very much, because it seems that you are interested only in the principal orbits, but your question was too tempting to not underline this construction.