I haven't commented much on this forum lately, but here goes a long-winded response:
I agree with previous comments that your plant needs a lot of work. It is the reason that you are often under. You need to keep your hands high and the pole centered so that you can control the pole drop without slowing down/striding under. I have some suggestions for drills that will help you fix it:
If you have shin splints, you should be doing about 30-50 walking plants every day , while focusing on keeping your bottom hand about 8 inches directly in front of your sternum (with your elbow cocked under the pole), and your top hand in line with your hip. Also focus on dorsiflexion of your feet while rolling onto the balls of your feet with each step...this will help strengthen your shins while helping you learn to run properly (and faster). Make sure that your hands are moving smoothly ahead of your feet, so that your top hand flips the pole just before the 3rd to last footstrike, the pole rises above your head just before the penultimate footstrike, and the plant is fully extended BEFORE your takeoff foot strikes the ground. The bottom hand should never drop from its position in front of your sternum. If you don't drop it, it will have only one place to go during the plant: straight up, and fast!
Also lots of 1/2 speed pole runs with the same points of emphasis, and an added element: don't let the tip of the pole hit the ground until you have left the ground. This will get you used to the feeling of taking off without pole support from the box. Hopefully these drills will get you a lot of benefit without beating up your shins. Remember that runway time often only reinforces bad habits! I always tell my vaulters to take advantage of an injury to do lots of quality drills and strengthen their fundamentals on the pole carry/drop/plant.
I disagree about your bottom arm; it actually looks pretty good, except during the plant motion. Pulling with the bottom arm is ok once you finish the takeoff by driving up and through the pole, as suggested previously. You may have some luck focusing on driving your chest to the back of the pit through the takeoff, which can help get the bottom hand up over your head while keeping the hips back momentarily. A good drill is to take a few steps and jump up to catch a highbar, trying to keep your chest forward and hips back. The key is the chest/knee drive and arms UP, not OUT. You can add the swing once you learn to keep the hips back.
Your swing needs a lot more work. Try learning how to punt a football with your left leg. You wouldn't want to kick the ball with a bent knee, would you? You would snap the knee joint straight at precisely the moment that it impacts the ball (this represents the chord of the pole).
You can get a much better swing by gluing your right foot to your butt at the takeoff, and then kicking your left leg straight through the chord of the pole while engaging your shoulder/core muscles and keeping your right knee fully flexed to the top of the swing. This will allow you to connect your leg swing to your torso, rolling your hips up higher and faster. You will notice that you get to the top of the pole WAY faster using this type of swing, keeping pressure on the pole, and you have the added benefit of being able to explosively extend your flexed right leg at the top of the swing, which helps you to get in line with the pole and spiral around it.
Watch this video of Bjorn Otto booming over 19' 1.25": his plant has a lot of the same issues that yours does, but his aggressive take-off, knee drive, and straight leg swing gets him to the top of the pole insanely fast so that he can catch the ride:
http://youtu.be/1iUkXSE73W0 (note to naysayers: I am not suggesting that Otto has the perfect vault, but he is a great example of the element of technique I am talking about)
Good luck, keep up the good work!
Tom