Excess saliva.

First of all I am very sorry to hear of your disease.

it is certainly possible for you to try different mouthpieces but I am afraid that the problem may require a more complex solution (try that first, , Personally I'd tyry a bigger external shape than your metal mouthpiece and probably a more closed one , in this way you would relax even more your enbouchure stimulating the salivary glands less).

If this fails, and may very well do, then you may consult with a specialist , the extrema ratio may be tone or more of these

https://www.parkinson.org/Understanding-Parkinsons/Symptoms/Movement-Symptoms/Drooling

but OP is, unfortunately for him, talking of a medical condition which exceeds the typical situation in which a player produces too much of a wet sound.

double post (seen many today, there must be some problem)

cannabis has been proven useful for many things but in this case botulin (also a natural product) shows more promises

https://www.cochranelibrary.com/cdsr/doi/10.1002/14651858.CD012408/full