I sometimes wake up with dry mouth, but usually that's when I've been tossing and turning a lot in my seep. If your mask does not have one, you might consider trying a mask with a chin strap.I have trained myself to breath through my nose, even though my mouth still feels like I am ready to spit cotton in the morning.
Strangely(?), the best cure for morning dry mouth is to brush my teeth in the AM.
When I first started using the CPAP, I did use a mask with a chin strap. Last year when I replaced the mask in use, I changed to one w/o the chin strap and have been fine for the most part. It took some adjustment fiddling and a little time to get used to the new mask, but it's fine now. If you are regularly experiencing dry mouth, that's an indication that you're getting pressurized air into your mouth, though not necessarily that your mouth is open. I would speak to your Dr about that, or visit a medical supply place local to you and take a look at alternative masks and see if you can reduce/eliminate that issue.