It's due to the sympathetic/parasympathetic nervous system. Sympathetic is for fight or flight.
Parasympathetic is rest and digest. When you run you are using the sympathetic nervous system. Parasympathetic shuts down so that all your energy goes to sympathetic. Saliva
is part of digestion. Parasympathetic (rest/digest) turns off, so less saliva giving you a think saliva feeling in your mouth.