If you call someone foul-mouthed, you are saying they swear too much — rude words come out of them as a habit, not as a one-time slip. It is a judging word. Parents, teachers, and newspapers use it; the person themselves would not usually smile and claim the label. It is about language, not about a dirty face or bad breath. A foul-mouthed comedian might be famous for it; a foul-mouthed boss in an office is a problem. Though actually, foul here is the old “morally dirty / disgusting” sense, the same foul as in foul language — not a sports foul (a broken rule), and not milk that has gone foul. The compound is plain once you hear it: a foul mouth, then made into an adjective with -ed.