Apnea sufferers do stop breathing through both their nose and mouth -- that's an apnea event occuring, you're not breathing. The next attempts to breathe, if you have Obstructive Sleep Apnea, will be through either nose and/or mouth, it depends on the person. It's all while you're asleep so you wouldn't know

Your CPAP will make sure you can get air through your nose if you need to, provided your pressure is correct and your mask is not leaking or you're not mouth breathing.
If you have a fully data capable machine you can check the display for your nightly AHI (the average number of apnea and hypopnea events the machine did not resolve) as well as your mask leak rate. A high leak rate with a nasal-only mask system is usually due to mouth breathing. (Not always, but that is the most common reason.) When a high AHI is coupled with a high leak rate, it is usually because the leaking is sufficient to cause loss of therapy pressure and more apneas/hypopneas occur. The machines can compensate for a fair bit of leakage but not 100%.
Some machines are compliance-only recording (e.g. how many hours you used it), so if you're not sure whether your machine tracks AHI and leak let us know the exact make and model, someone here can tell you whether you can find this information.
If you don't have a fully data capable machine then you may have to experiment with your full face mask or ways to keep your lips sealed. (Search this forum using the search bar above, or the google-style search in the upper right. Lots of ways have been posted, to deal with sealing lips.) Or ask your doctor about this. It is possible he/she could get you a loan of a fully data capable machine for a few weeks, to find out what is going on.
Blessings,
--pseudonym