I've dived a Sherwood Oasis my entire diving life (first certified in 1986, DM in 2000; quit logging but close to 1,000 dives). Whether it works or is a placebo (I believe folks will come down on either side), I have never experienced dry mouth while on a dive.
If your issue is significant to where it is a distraction, the Oasis regulator may deserve investigation.
I second this. Since diving a Sherwood Oasis I have never had an instance of dry mouth. Its never even a thought.
The Oasis was gifted to me and it was after noticing the difference I looked it up and realised that was what it was marketed to do and went, "Oh, 'Oasis'... I get it."
I literally have to swallow saliva sometimes during a dive.