Let's assume that classical logic refers mainly to the law of excluded middle, more precisely a bivalent logic. Let's assume that Modal logic refers mainly to the situation were there is a (loosely specified) universe with worlds and a reachability relation between these worlds, and that we are in one specific (well specified) world of this universe and talk about propositions in our own world and the (relatively well specified) worlds that are reachable from our own world (and perhaps also the worlds reachable from worlds reachable from our own world, and finite iterations of this construction).
Let's focus on propositional logic to simplify things. In classical logic, some natural language sentences correspond to propositions, and every proposition is either true or false. In classical logic, you better resist the temptation to assign a proposition to every natural language sentence, because the negation of an ambiguous sentence might still be ambiguous, and hence the functoriality of logic might fail.
One of the problems with a classical logic account of Modal logic is that there can be natural language sentences which correspond to propositions in one world, but fail to correspond to propositions in another world. One idea to remedy this situation is to only consider natural language sentences which correspond to propositions in every world of the universe. But how much will you restrict the expressive power and the useful applications of Modal logic by doing so?