I had this exact issue with a pair of students this weekend in the pool. One would constantly exhale through his nose (which seems to be the opposite of your concern), and I worried that he would always have a leaky mask from breaking the seal with his exhaled nasal breath. He had no problem breathing with the mask off, as I constantly saw exhaled bubbles from his nose after inhaling through the regulator with his mask off.
My other student of the pair seemed to experience exactly what you describe. He had no problem breathing through the regulator, but when it came to clearing a flooded mask or breathing without one, he would take four or five breaths to accomplish it, as only a tiny spurt of air would come out his nose as he attempted to clear.
What we started with was on land: just a repetitive cycle of inhaling ONLY through the mouth (like sucking through a straw) and exhaling ONLY through the nose (like blowing into a tissue when you have a cold). Over and over and over we practiced, until the routine became embedded in his muscle memory.
Once back in the pool, the same issue returned, because the presence of water near your nose is a powerful inhibitor of opening that passage, even to EXHALE. But using the same mindset as we had at the surface, and coaching him to "blow his nose into a tissue", he reacquired the habit he had shown so easily on land.
Your problem is VERY common, and is merely a matter of teaching new individual muscles to do something they are disinclined to do. With repetition (just like in grade school with your ABC's, over and over), what is difficult becomes easy.
I always start my pool classes asking my students if they can flex their fourth finger BY ITSELF. We can all flex our index finger, or our thumb, but it takes a bit of effort and repetition to find the nerve connection that tells just one muscle to act in a fashion different from before. With that analogy, and making a game of isolating the mouth from the nose, we can almost always get past the sort of issue you have described.
Just keep practicing at it, and don't lose hope! It takes years to play a violin. At least scuba is easier, if not less daunting when you first start. You will do just fine!