Screen protector? How retro.
Test if it's a hardware problem by enabling "Pointer Location" in Developer Options. If you drag your finger across the display it will continue to draw a trail behind it. If the digitizer loses your finger the x and/or y coordinate will light up red. If you retouch the display it will start drawing a new trail. So if there is an issue with detection you should see the current trail disappear and a new one start even though you didn't lift your finger off of the display. I wouldn't stress taking the screen protector off. I've been using my S3 without one for two years and I don't have a single scratch. The find 7/7a has Gorilla Glass 3 on it so just don't wipe it with a cloth or shirt that has sand on it and you'll be fine.