I believe the best drill of all drills to do is what I call "runovers"; probably numerous other names for it, but is what I call it. Simply, one vaults on the flat ground and swings through to the take off foot. With runovers, one has to perform a free take off. Well, maybe not every time, but he or she might get tired of falling to the ground. The grip starts at a point where an individual can jump up from a standing position and touch the pole when it is resting on the ground in a vertical position. This is a starting point. Then, I suggest using just 4 steps (R-L-R-L for a left foot take off), carrying the pole to the side as if one was vaulting, bringing the hands up, placing the tip on the ground and swinging through in a "figure 4" position landing on the take off foot. And remember as you swing through, keep that top arm straight. Is this too elementary? Not at all! If done correctly in a straight line, you've done a free take off. There is however, no bumping the pole with the trail leg! Because, there is no bumping of the trail leg when vaulting! Now, as one gets better at the drill, move the top hand up. How can you tell if you're getting better? If when you land and your speed is good, move the hand up. The objective is get the hand up (progressively) the pole as high as one can while still able to move the pole forward. Now, because you can move the hand up you are getting better at a free take off. More Importantly, you're learning to jump up a take off.
Which brings me to the problem of being "under". Pole vaulting is a jumping event. One must learn to jump just as in the high jump or long jump; take off angles are just a little different. So, the runovers help teach you a better jumping take off. Many vaulters "run through" the take off instead of jumping up on the pole.
After the ground runovers, go to the pit using the same 4 steps only now raise your hand about 8-10" up the pole (8" for the depth of the box). And, if you trully want to vault better, don't be lazy or cut corners. In this instance, have a tape measure set along the runway starting at about 6-7' from the back of the box. Now, determine the distance from your projected take off point at 4 steps and run back; many are at 20-22'. Turn to the box, run 4 steps and essentially, do a "runover" into the pit. Nothing is different from the ground runovers except your hand is higher up the pole. Oh, and should you go back to 6 total steps, nothing is different from the runovers on the ground, 4 steps or 6 steps except you will hold a little higher on the pole and run a little faster. But again, have a consistent starting point and take off point for 6 steps also. If you don't have a consistent take off point, ask yourself: "am I jumping up at take off". And with either 4 steps or 6, continually move your hand up the pole challenging and improving the jump.
You should be able to execute a reasonably good free take off holding up to a hand grip of 12-6 or 13'. But don't fret, there are soooo few high school vaulters that vault above their hand grip and are able to clear heights of 13'-14'. what someone like you needs to do is work on your approach and developing a good, high hand take off where you're jumping up on the pole. Once you get more accomplished at the approach and take off, the swing will progressively follow. but, one needs to get a lot of good reps
Sorry for being so long winded, but trying to help someone who seems to be eager to get better. There is no magic and requires a lot of work. Good luck and be smart which will keep you safe.