Here are a couple of points that I'd focus on to improve your skiing.
Width of stance is fine, don't try to widen you stance, but work on your balance. Remember that a wider stance = better stability, but a little narrower = more agility. Therefore, narrower stance for slalom-type turns is fine ... as long as you're balanced.
You should try one-ski balancing exercises, which are a challenge and can just be plain fun. Try skiing on the outside ski, lifting the inside ski up a little - make sure you are balanced throughout the turn (or as much as possible) on the outside ski - don't cheat or fake it. This could take awhile to perfect, but keep trying.
In order to balance on the outside ski properly you'll need to use counter-rotation (counter balance). THis means make sure you're shoulders and hips are pointed down the slope and NOT too square on your skis.
You'll also need to bend (angulate) to get your upper body more over your outside ski.
Balance is the key to great skiing - for more of a challenge try skiing on only the inside ski
Oops ... I guess I should say what I see in your video: The main things is as others have mentioned, (1) the sliding of the tails and (2) in long turns you often have too much weight on the inside ski. Doesn't hurt too much on gentle terrain like the video, but on steeper stuff it's gonna be trouble.
That's my two cents and thanks for welcoming new people to your forum.