Its a bit hard to give a proper answer as you are at the beginning so the reasons it happens might be a few different ones
Since from other information in the comments you say that this happened during the pool sessions of your course and you are not certified yet, I don't think is ok giving you too many tips about what to do, or suggest you to use one technique instead of the other. Its your instructor the one that has the responsibility for you and (for the moment) you should follow their methods and refer to them for resolving those things. Other suggestions might conflict with what/how they are teaching and that would be annoying for them and for sure not good for you.
Masks problems are very annoying and frustrating, the instructors are there to help and will take the time to solve it for you, don't be afraid of asking them.
The important thing is that if the problem persists when the open water part of the course comes, you don't freak out, you will be fine. Ask your instructor if they can keep an eye on you and if they can stay right beside following you during the descent and correct you if necessary. (Don't feel weird about it, that's the reason you are required to take a course to get certified and they are there to fix these things).
from the comments you said you tried only the Valsalva. As it might be related to the way you perform the maneuver rather than the breathing in itself, be aware that there are other equalization techniques and you might prefer one of those. Given your situation I wont list them, they are described in scuba articles all over the net and you can discuss them with your instructors (don't use them without having talked with your instructors first). However don't be lazy and change technique just because one doesn't come out right, its a good thing to try to perfect all of them, even if one works better than the other now things might change in the future, and your instructor might want you to stick with the Valsalva.
You should also have learned to equalize the mask so, in itself, breathing through the nose into the mask shouldn't be a disaster like you describe. Strap and pressure should keep it in place.
This brings to the mask fitting:
You said in the comments that you had the two masks you tried fitted at a "reputable shop". Be aware that its not if the shop is reputable or not, the fact is that fitting dry just goes so far. Some masks simply don't fit well some faces (although a leak is different than a mask just leaving your face). Even the strap should be properly regulated and positioned.
Like many divers I went through quite a few masks before finding the perfect one, they might feel and look all similar but a small detail can make a big difference.
I would suggest trying all the different models you can, and I mean different shapes, amount of coverage of the face, with and without purge valve, different volumes etc. (Lower profile masks have less volume, some like them some don't). Your dive shop could have some demos, tell your instructors you want to try all you can beforehand so they can bring the demos at the open water sessions (this means buying the good one from that diveshop too if they go to all the trouble to help you, some are very expensive but don't go look for the best price online).