I will leave the mask comments to those with more overall knowledge/experience than I..
But you may benefit from training yourself out of being a mouth breather as well as looking at the other aspects.
There are threads on the topic.. but the aim is basically to get your tongue to seal against the roof of your mouth and/or your throat muscles to close your gastric tract so that air only comes in and out through your nose when sleeping. (sorry for the non biologically specific description)
One technique to practice is holding water or other liquid in your mouth and breathing through your nose while doing so.
Also consciously practice placing your tongue against the roof of your mouth and keeping your mouth closed while breathing in and out at times during the day.
When I first started CPAP a couple of years ago I found I would sometimes wake in the middle of the night with a dry sore throat.
It was a random thing, so it took me a couple of months to figure out that my mouth was dropping open at times when the muscles fully relaxed in deep sleep, it wasn't until I came fully awake one night with air rushing out my mouth that I realised I was even opening it in my sleep.
( I wonder how many bugs and spiders have walked in there over the decades
)
I have not had a problem with it for ages now, I use a Fisher & Paykel Eson nose mask as my sensitive nasal lining can't tolerate nose pillows.