How can you tell if the stage at a bluegrass concert is level?
-the banjo player is drooling from both sides of their mouth.
Those irritated by grain of sand best avoid beach.
-Strumelia proverb c.1990
"zen dulcimer". Ah. Grasshopper must learn seal lips when play music! When I do deep meditation and really relax, my lips, teeth, jaws, etc. I can have the same issue. Luckily I meditate that deeply when lying on my back, so the liquid goes in not out. Try playing dulcimer laying on your back -- strap the instrument to your music stand
This is my recommendation. Record yourself playing in DAA. Listen to it every time you eat really good food. You will slowly train yourself to drool equally out of both sides of your mouth. Get yourself a bib and the problem is solved!
But at least you have your stage name: Drooly Don Grundy. It has a ring to it, doesn't it? I would pay to see a band called Drooly Don Grundy and the Doughnut Holes, wouldn't you?
I heard a radio conversation with a physicist whose wife was a cellist. He said that when he is working hard, his right arm pit sweats more than his left arm pit. But when his wife is working hard, it is the other way around. The theory is that he was using the more analytic side of the brain, which controls the right side of the body, and his wife was using more the creative side of the brain, which controls the left side of the body. Maybe you are experiencing something similar.
Actually, I suspect Strumelia is correct. Because you are concentrating on what you're doing, you are not swallowing as often as you normally would. I have the bad habit of holding my breath for really tough parts of a tune, and then you can hear me exhale deeply right afterwards. I've had to delete a lot of recordings for that reason.
Probably a sign of intense concentration in the music. Lots of musicians do weird things with their mouths and their faces while playing, btw. I suspect that's why oldtime fiddlers like to sport beards so much.
If you can remind yourself periodically to close your mouth it won't matter if you're drooling.