So, just to make sure I'm understanding this correctly... the problem you're seeing doesn't occur if you use your S-pen to type? There are two digitizers on the Note series... one for touch, and the other for the S-pen. If you can use your S-pen to type and not your fingers, then your digitizer (touch) may be going bad. It really sounds like a hardware problem considering it only happens after the phone has been used for a while. Sounds heat related.
The reason I asked about landscape was to find out if the bottom area of your screen was failing. Say, for instance, one of the intermittent letters was the letter "a".... if you put your phone in landscape, the letter "a" would be way to the left of the screen (away from the bottom while in portrait). Then again, I usually ask this when I suspect the user might have a case on the phone where a magnetic closure is involved... which is normally near the bottom of the screen in portrait mode. Might be irrelevant testing since you say your case has no magnets.
As for screen protectors... On my Note 10.1 2014, I had to remove my tempered glass screen protector because it was causing a lot of intermittent touch control. Then again, tablet screen protectors are rather thick. On my Note 4, I use an iCarez tempered glass protector with no issues. It's only .15mm thick. I wouldn't think your screen protector is causing the problem if, like you say, it's fine until the phone warms up. Of course, perhaps it's normal for the screen to become less sensitive if it's warmer, and in your case having the screen protector on makes matters worse. How thick and what brand protector do you have on? Manufacture/model number? I would really hate for you to take off the protector to test only to still have it fail without it. Then again, it would be cheaper than a new touch digitizer.