First of all, those are 30-degree slopes. The 45-degree slopes are so steep you slide down them if you try to stand on them.
The issue you see isn't angle jittering, it's caused by the demo having "interpolate colors between pixels" turned off, so all graphics are snapped to pixel-perfect coordinates. It usually makes pixelart look better, but having slopes and rotations gets all the sprites into non-integer-coordinate territory and then things get a bit iffier.
Your two main options to fix this would be either having interpolation turned on (which works the best if your graphics don't have a lot of crisp pixels), or have a special "draw angle" for the player that is snapped to, say, the closest multiple of 15 degrees:
drawangle = (15*((image_angle+7.5) div 15)) mod 360
Then just draw the sprite with that angle.