Yes, you are fixing it correctly. I see you already discovered that she is dropping that inside shoulder.
Hip control, moving the hip into the lead is important, but at the same time you have to make sure that inside shoulder is up and the horse is not leaning on it.
Holding that inside rein up against that inside shoulder will give you that shoulder control
Also, a green horse will often lean on that inside shoulder, and if they do, I will first fix that before asking for the lope departure. I do this by bumping them around on a turn on the haunches in the opposite direction of the lead you want, to get them off of that inside shoulder, then set them up again for the lead departure
Hope you had a good show, as I got to this post late