She's a really really cute horse! I like her movement a lot. I think to help with your balance you need to shorten your reins and carry your hands more in front of you. You don't have to take up more contact, just keep your hands in front of the withers and shorten your reins accordingly. This will keep you from getting behind the motion; you tend to fall backwards a little. Staying with the center of gravity will also really help you stay secure during her fits. Make sure you keep your thumbs up too! I saw a lot of puppy paws. And I also think you should shorten your stirrups. I know you're in a dressage saddle but with a green horse I think shorter stirrups add a lot of extra security- they help you get further down in your heel instead of reaching for the stirrup.
As far as helping her; I think you're doing a good job. Just make sure that when you're working on steering, think about using your hands as a last resort. First look with your eyes, turn with your body, use your legs, then
use your hands.