Would dry mouth occur with a concussion?

Yes. All concussions need a neuralogical consult. The sympathetic ANS parasympathetic nervous styems originate in the mid brain. Loss of salivation is a neurolgic deficit, nad may indicate mid brain damage.
Doubtful. In all reality, while it is possible, there are many more causes for dry mouth. You should see your local md for evaluation as concussion is highly unlikely to cause dry mouth. You should be evaluated for another disorder like sjogren's syndrome.