Interesting you find air holding breath, but not when lightly exhale. I suspect the slight exhale is triggering the head to push down slightly. The difference between getting air or sucking in water is about an inch, doesn't take much of an error in head position.
Re: Over-rotation. Meaning one shoulder is stacked on top of the other, body is rotated 90 degs and is unstable and sinks. Body rotation should be between 40-60 degs, one shoulder just clears surface.
I didn't realize you are not swimming, but only doing drills. It's time to take a few strokes, don't be stuck in endless drill mode. If you are doing swing skate/switch, sweet spot - you are able to take 8-10 strokes and swim single lengths. Take four to six strokes without rolling to breath, repeat. When you've acquired some stroke symmetry, then integrate breathing. Snorkel would be fine to allow getting air until you have establish stroke symmetry and rhythm.