Is it normal to have a watery mouth when using a tongue thrust appliance?

Yes: Any time an appliance is placed in the mouth, the mouth salivates more because it thinks it is food that needs digesting. After the appliance has been in for a while, the body gets use to it as not food and stops the excess saliva flow.

Watery mouth: If you mean by watery mouth that there is a lot of saliva then the answer is yes. If the appliance is new or relatively new then the body may think the appliance is food and is increasing up the saliva production. What can do done is to swallow more often. Eventually the production should return to normal.

Very normal: The body automatically senses when something new and different is placed in the mouth which then triggers the salivary glands to produce more saliva. It's also normal for an individual to start running his or her tongue around and swallow differently. This triggers the same reflexive production of saliva. This tends to subside over time.