Ok. First of all, that mare's a KEEPER!!! She is a dandy horse and will go far, I tell you. And if you dont' want her, ship her to me , pronto!
YOu handled her nicely on the ground, just that you swung the saddle up with the fenders swinging up over her wither, rather than them over her haunches, which is more typical. Might be best to do it the "normal" way. And when you want her to move over, try using your body language first, without actually touching, and then go to touching, if necessary. You many not need to touch. However, if she swings into you suddenly, then of course, like you did, you push on her pretty quickly and firmly to get her away from you.
As for the riding, I can see right away that you riding too much on your "fork". Have you trained in hunter? You ride a bit like a hunt seat rider. You lean too far forward, have too much arch in your lower back (thus are kind of on your pubic bone) and your upper body is stiff, with your arms too straigth (no bend in the elbow) , angled downward too much and too locked.
This position means you will always be prone to tip forward, if the horse stops suddenly, and it is pretty hard to gather a horse to collection if you are leaning onto its' forhand.
I think that mare is not moving forward as nicely as she is capable of because she feels your position as being a bit unstable and tipped forward.
Sitting in the saddle (horse standing still) , lift your feet up and out of the stirrups and up onto the horse's withers in front of you. YOu should now feel your seatbones very firmly in the saddle. Sit up really tall, no curling forward!
Then, drop the legs down ; just let them fall as far down as they will go.
Now, without changing anything else, lift up the front of your foot and catch the stirrup and put your foot where it should go.
Feel the seat bones still? Sit up really tall and ride forward, elbow bent, riding like a proud indian princess and with your chest up and your sternum being the leader.