Alrighty then. I've had a chance to look at your video (as brief as it is).
To answer your question about throwing across your body, it does appear that you stride to the closed side a bit. (But it's kind of hard to tell from the camera angle.)
If you want to make an adjustment related to this, I would recommend moving your starting position to the left side of the rubber. To figure out how far over to move, throw a few pitches as you normally would off a dirt mound and look at where your back foot drag line ends - it should be off to the right side of the rubber centerline. How ever far off the centerline of the rubber your drag line ends is how far over to the left you need to move. You want your drag line to end on the centerline of the rubber. This adjustment will cause you to stride onto the centerline of the rubber instead of away from it. As a result, you will still stride to the closed side a bit but when you "turn the corner" to square with the target you won't have to turn as much. In other words, you won't throw across your body as much. This is a fairly benign adjustment to make in that it you're not actively trying to do something different. If this doesn't correct things enough, then you will need to consider changing the direction you stride.
I would also suggest that you try getting the hips started forward sooner (while maintaining the same knee lift) to build more momentum down the hill. The added momentum will let you put more zip on the ball using your body.