SkiMango I would encourage you to not let your hips square up to your skis, but instead have your center of mass (CoM) facing towards the apex of the next turn. What you described sounded like you were starting each turn with a upper body move to create counter and then having your hips and then legs follow.
What the animation shows is a stable upper body and the legs turning more then the hips. This specific animation is of a pivot slip and thus the extreme of what we are talking about. in more 'normal' skiing you would stay in the corridor nor side slip as much, but the concept is the same.
Your body want to be completely aligned. That is the position you would take if I asked you to stand in one spot for awhile. By having you CoM facing towards the next apex you create a twist and tension. When you transition into the new turn you release the tension, you body untwists and then you continue to turn your legs across the fall line creating twist and tension on the opposite side (skiing into counter).
What you describe, you allow the twist to unwind before you start the turn and so you have to recreate the twist by turning your upper body first. Essentially you are adding an extra unnecessary step.