Neurologist or Pulmonologist?

The place to start is with your original sleep doctor, the sleep physician treating your osa.

For a 2nd opinion, it would be best to go to a sleep physician who was also a psychiatrist.  If such a doctor isn't available, I would recommend a neurologist/sleep doctor.

I am using the same machine as the one you are using. I haven't experienced the drooling but when I sleep on my back I have awakened with the air getting into my mouth. The air pressure seems stronger than usual. I'm still getting used to the whole process. Despite all the problems, are you feeling better during the day? I read someone say they would wake up feeling refreshed. I did have a day that my energy stayed for the entire day. One day, amazing, one day in I don't remember. I hope it comes around again, sooner.

I'm a neurologist with OSA. In my opinion it makes no difference whether you see a neurologist or a pulmonologist as long as they are board certified in Sleep Medicine and their sleep lab and technologists are certified.