In my opinion, it happens because when you are sad, you are, at the same time, less confident about you, no matter the reason you are sad. When we are sad, we become more vulnerable and wavering on things we do, it's as if we can't concentrate ourselves properly ir our actions, Idk. Our mind is too focused on the reason of sadness, so things can be disordered in this moment. But objectively answering your question, I don't think sadness alone can be a real cause of stuttering. As I said, it can increases stuttering on one who already have It, but not be the cause of It. Wish you all the best, my friend!