I do not know if you have any physiological problems, but the environment which you play may be the problem. Blowing warm moist air into a relatively cool horn creates condensation on the inside surfaces. Lowering the humidity of the room in which you are playing may help, but if this done by air conditioning, you are getting a colder environment and this can lead to a cooling horn and more condensation. I am guessing that most of the water in the horn is from condensation on the inner surfaces. I wonder if lower notes produce have more warm, moist air than higher.

If you suspect that you are producing more saliva and you need to swallow a lot, you may want to check with a physician. It seems the larger complaint is dry mouth and medicines that increase dry mouth.

I wonder if our more open and relaxed playing style could tend to have anything to do with it.

These are just my thoughts on the subject.

Experiment with where you play, keeping all other conditions the same (what you are playing, how loud, etc… and see if there is a change.