You do need to turn more with your skis. Tipping of the skis and angulation will help you achieve a balanced position over your feet, both fore and aft as well as side to side. This will make for more comfortable and effecient turns.
Let your skis go flat in the transition between turns to release the old turn, then tip your skis onto their new edges into the new turn. When you let your skis go flat, they will automatically start to turn down the hill on their own. Tipping will get your skis on edge so you can control the turn. Tipping both skis will move both knees to the inside of the turn and your knees will stay about as far apart as your feet are apart. Think of tipping as putting pressure on the inside foot's little toe and on the outside foot's big toe ('big toe, little toe'). You want to make your foot, ankle and boot roll to the inside of the turn to put the skis on their inside edges. In the videos, you are twisting your skis around to start your turn. When your skis go flat in transition they automatically start to turn downhill. Tipping will get both skis on edge so that you can control your turn with both skis.
Your fore/aft balance needs to stay over your feet. Stay in an athletic position with your ankle, knees and hips flexed and supple. There are times in the video that your outside leg is straight and tipped back. This is blocking your body from being able to flex as well as moving your weight back. Maintaining an athletic stance help you keep weight forward so that you can pressure the middle and the front of the ski; where it is most effective. To find your athletic stance jump up and land. Your feet will be a natural distance apart and when you land your ankles, knees and hips will flex to land you gently. That flexed position is your athletic stance.
Lateral balance needs to be focused over your outside ski. As you pass through the transition you should be standing on both skis fairly equally. The pressure on the outside ski should increase gradually as the pressure on the inside ski decreases as you turn. As you approach the transition to the next turn, the pressure on the outside should decrease until you are standing on both skis equally again. The video shows that your inside leg is supporting you more than the outside leg in many turns; this is an ineffective position to be in. As you learn to tip your skis to start your turn, your knees and your hips will naturally move to the inside of the turn. To compensate and maintain pressure on the outside ski, you need to keep your upper body slightly towards the outside of the turn. This will mean you bend a little sideways (toward the outside of the turn) or angulate.
There are some exercises that might help you. I'll suggest a couple. Start with putting your hands on your knees while you ski. Make sure that you tip to start your turn and that you feel, with your hands, both knees moving to the inside of the turn. You can try this at home, too. Experiment with tipping, so that your 'outside' edges come off the floor while your knees move to the inside of the turn. Keep your shoulders over your feet so that the pressure on the 'inside' foot doesn't increase. When you do this exercise on the snow, the force of the turn will be pushing you to the outside ski, so the pressure on the outside ski will become greater than on the inside ski.
Once you are comfortable tipping both of your skis to turn, try just touching the outside knee during each turn. As you turn to the left, touch the right knee, then as you turn back to the right, touch the left knee. Make your touches smooth and progressive, so that the motion to move your hand to your knee and back to its normal position takes the whole turn. You will move your upper body to the outside of the turn while doing this which will result in exagerated angulation. This will get you used to the feeling of your upper body moving to the outside of the turn while your hips move to the inside.
When you are comfortable these exercises, you will be turning using tipping and angulation.