I think it may depend on how many dives you have done.
The reason I say that is because I posted a question regarding dry mouth.
When I had 1 to 50 dives I got dry mouth a lot and it irritated me much.
The irritation seemed to be forgotten after ~100 dives. Some divers seemed to have the same experience of "growing out" of dry mouth issues.
After all those dives, I believe I learned to periodically press my tongue to the roof of my mouth and gulp spit.
Also, as I got more experience, my breathing became deeper and slower so it likely also contributed to less irritation.
I used to run Aqualung Conshelf 21 type second stages which is a simple downstream design. The venturi assist inside is non-adjustable and diverts the air from a small vane. Maybe this directs a cold focused shot of air to my throat.
I now run barrel-poppet/pneumatically-balanced seconds (Hollis 212, Aqualung Legend and Apeks ATX-XTX seconds) that may provide a more diffuse blast of air. I tried the Apollo BioFilter and hated it. You need to "charge" the media with water every dive and depending on where you mount it the device is another point of easy damage. It is easy to accidentally grab it or bang the long cylinder hanging off the threaded fittings.
Maybe borrow or swap regulators with a buddy and see if it works better for you.
A not so bad advice to cure the issue it may be to get to get at least 100 dives soon (Unless you already have that many. Heck, get another 100 dives to be certain.)
At one point, I carried a "bag" type fruit drink (Capri sun) having foil containers. I could take a nice drink underwater after punching the straw through the bag. It was really a novelty rather than gear I needed to have. It was fun to say I did it before.