@Wendy, I have a skier’s edge. Yes, it can help your skiing. As it comes over the top, press harder with your outside foot and lighten (not pick up) your inside foot.
Another exercise for you to try. As you move your legs from side to side try not to let your hips and upper body rise. (I call it keeping my eyeballs level.). You have to flex and extend your legs to do it. This will give you the feeling of a cross under move. It also gets your legs extending and flexing. That learning of extension and flexion will improve your overall skiing. Start with little movement side to side and increase your range as you become more proficient in the move and feeling generated by it.
I'd be careful about the skier's edge. One of the issues in your skiing is that you are pushing your outside ski away from your body. The Skier's Edge requires a push to extend the outside leg. IMHO, your issue is twofold: you don't know how to tip your lower legs (so you incline into the turn winding up on your inside leg) and you push the outside ski away from you in the late shaping/finish of the turn.
So, if you have a Skier's Edge, think of using it primarily to get in shape. Also pay attention to early transfer of weight to the new outside ski. But be careful about the extension.
When you get on snow, I think you need to focus on lower leg tipping. Take the leg extension out of the equation for a (good) while. You need to learn about tipping to establish the (outside) edge and platform and then allowing the ski to turn. You are looking for the ski to do its thing, rather than forcing the ski to do something. You want the snow running along the length of the ski, not across its width.
Here's a video of Andreas Spettel on edging. Practice some, or all, of these things.
Outside ski drills, where you pick up the old outside ski before edge change, will integrate the outside ski to outside ski approach identified above with lower leg tipping. But you first need to work on the tipping exercises, IMO.