Okay, so, I managed to beat the demo (after many deaths on the moving platform-aura room) and I think I figured out what was tripping me up so much.
In that one particular room, there are lots of little platforms scattered around and along the bottom of the stage. Because of this, I thought I had to go under everything, which ended up causing me to die a whole lot, when really the easier path is to go above everything. Also, there aren't any checkpoints in that room, so any mistake was effectively a reset (although the room was fairly small, so it wasn't too bad once I figured out what to do).
The controls feel much better by the way! Thank you for taking my thoughts into account!