Carpet burns, ouch!
The edges will catch on real snow too, but snow may be more forgiving. Remember pressure control as you move ski to ski, if your COM is over it when you tip, it’s going to engage to a greater extent than you may want or be prepared for, It will take some time/practice to figure that pressure transfer out.
Don’t worry about the wedge, even if you are doing just that you are building some muscle memory and more importantly muscular endurance/strength. Of course you want to move past the wedge, but at the same time you can also establish a very strong/stable platform. Consider trying to move into a Wedge Christy, where you are parallel skiing through part of the turn, guiding the inside ski into a parallel relationship along the traverse.
No recommendations for the carpet burns though!