My guess is they started this before talking movies/cartoons in an over exaggeration to convey to the audience that the character was nervous.

“Humans swallow when nervous. The act of swallowing helps to clear the throat and maintain normal function. So, whether we’re excited, worried, or nervous, our bodies continue to perform essential functions like swallowing without conscious effort.”

Seems natural to extend that to cartoon characters, whether human or animal.

It’s a pretty normal response, similar to taking a deep breath before something major happens like a job interview or a date.

Sometimes the mouth creates a lot of saliva, for the reason I don’t know. Better to swallow than accidentally spit on someone while taking.

@Blackberry, I believe that the saliva is from the fight or flight response.