It is good to hear that you are getting some help and some choices in masks. It is probably the most critical component of the CPAP treatment. Believe it or not, with all the masks and names, ResMed does have some logic to at least some of them. In general Full Face masks have a "F" in the name, Nasal mask a "N", and Nasal Pillow masks a "P". Not always but most. For that reason I suspect your nasal mask is likely the AirFit N20 not F20?. See link below:
AirFit N20
If so it looks like a well made mask with very good headgear. The issue with it, and the full face F20 is that while the headgear is substantial and should hold the mask in place, the mask itself sticks out quite a bit from the face and is still quite susceptible to being moved, and unsealed. I find that the AirFit P10 nasal pillow mask, see link below, to have a much lower profile on the face and is less susceptible to moving. You don't mention what your pressure is, but I have used it up to 14 cm or so, and it does not leak from pressure at that point. ResMed rates it as suitable up to 18 cm. The headgear on it is not the greatest, but it may work for you. I find wearing a chin strap helps keep it in place.
AirFit P10
SnuzyQ, a contributer here, has posted that she and her husband use the ResMed Swift FX mask. Despite the letters it is a nasal pillow mask, and quite similar to the P10. but with a more substantial headgear setup. It may have a slightly higher profile than the P10, so hard to say which may be better to address your stay in place issue.
My suggestion is to try and convince your provider to let you try both the P10 and Swift FX to see which one works the best. As for letting air out of your mouth, you can try a chin strap. What I have ended up doing is using both a chin strap and paper medical tape (3M millipore) over my mouth. It may seem surprising but I have found that using tape on my mouth to be the least invasive/annoying component of my CPAP setup.
One last thing is the setting in your CPAP for the different masks. You should be sure to ask your provider to set up your machine so you can select the correct mask type as you switch them around. Assuming it is a ResMed CPAP here is a link to a chart showing the correct setup for each machine and mask combination. Surprisingly the N20 mask should be set to pillows, not nasal.
ResMed Mask Setup