Althought you should try to overcome the problem without pinching your nose there is no requirement that states you can't pinch off your nose while removing your mask. I myself was fine for the mask clearing exercise but couldn't breath off the regulator for more than 2 breaths without the pinch. During my pool work my instructor had us lead around by our buddy with the mask off and I completed this exercise by pinching. While trying to overcome this I learned that you can have a coughing fit (or puke) while breathing at depth just remember to keep the reg in your mouth and cough right through it.
I would probably advise against pinching to do this exercise as it might form a habit where you can only do the exercise when pinching your nose. When you get your mask kicked off under water, it is unexpected so you usually don't get to prepare yourself for pinching your nose.
Anyway, I had the exact same problem and very nearly quit myself. I figured out that I breathe through my nose and mouth at the same time and just had to train myself to avoid doing this, perhaps this is your issue? I put my mask on in the shower filled with water, and I also snorkeled in my bathroom sink (the bath isn't big enough for snorkeling) without my mask on for 5-10minutes every day for a week. My partner found it hilarious but it did cure my habit of breathing in my nose at the same time as my mouth. If you try this, do it in cold water, as it is harder, and will help you learn better.
Another thing I heard is that tilt your chin forward when you have to clear your mask, rather than up like a lot of people teach, as if you look up you will have the water run down your nose and start to choke you. Like of make an elephant blowing its truck type gesture - ie. look down, remove your mask, replace your mask, inhale through the regulator, break the seal on your mask and as you start the exhale to clear your mask, move your head to look up to help get the water out.
Anyway, I was sick with nerves on the final day of my OW having to do the exercise at 18m as I had struggled the whole way through but with all my practice I ended up finding it very simple, and have no issues these days with clearing my mask (other than an icecream headache from the shock of cold water!), which is lucky as it has been kicked off or dislodged a few times already, either by other divers or currents. I have accidentally breathed in a few times through my nose doing them, but it is no biggy and you can just cough the water out through your regulator.
Good luck! Lots of people have this problem at the start so I am sure you can overcome it.