I am so sorry you are experiencing claustrophobia. Others here will likely have a variety of solutions, but I will tell you of my experience and what was recommended to me that worked, in the hopes some of it might help you.
I went through 5 different masks before I found the one that fit. I tried two different nose only ones, one that was kind of like a cannula and two full nose/mouth ones. I tried a chin strap to keep my jaw shut while using the nasal masks, but it just didn't work.
It also took a while to get the right air pressure- when the air wasn't going into my nose and rushing out of my mouth or being swallowed by me in my sleep, it was creating painful pressure in my nasal passages and giving me earaches. It helps to have someone who knows what to set it at work with you to make it as comfortable as possible. Laurie (Igelb) is a master at this and provided me invaluable advice.
The one I currently use sits over my mouth and nose- it's "comfortable", but intrusive. It took a while to get used to it. I was cinching it too tight, afraid it would leak air- it was painful and left bruising on my cheeks and nose. Once I loosened it a bit, it felt less intrusive and didn't leak as I'd feared.
I started by using it only while I was awake, and only for 10 or 15 minutes at a time when relaxing in my lounger. Once I got over the crawling "get this thing off of me" feeling, I started using it at night. I'll be honest, I take some really excellent drugs to reduce spasticity in order to get me to sleep, and to remain asleep the night through- so once I'm out, I won't wake up for 7 to 8 hours. At which point the mask gets removed immediately. While it is intrusive still, I don't wake up feeling totally wretched, foggy and with a headache the way I did before getting the breather.
It's definitely better to get used to it before you absolutely have to use it. Another completely unexpected bonus to the thing was my skin is in better condition than it has been in years. I guess ample breathing and appropriate levels of oxygen really is good for your skin!
Best of luck. You may never like the feel of it, but with time you can learn to tolerate it. The benefits of using one really do outweigh the discomforts.