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Why not try an EggBut Slow Twist Snaffle or even a CorkScrew bit?
I've heard those are PERFECT for hard mouthed horses.

If not, try a lighter, more comfortable bit. Like a Plastic Bar Bit.

Unfortunately, the humbo jumbo "stuff" you don't want to hear might fix this horse's problem.

The main purpose behind a FULL CHEEK is so that the bit doesn't run thru the horse's mouth if you have to pull their head around for some (?) reason. Changing over to an egg butt or d ring isn't going to do much of anything.

I would try the Kimberwick (Uxter) or Pelham if I had taken many lessons and got an honest opinion from someone on the ground that could give advise on body position/hands/etc. If the horse is pointing it's nose to the sky as she is running off with you she may be responding to the bit hitting the top of her mouth. Some horses just don't work well in a traditional "snaffle" bit.

The general rule for any bit of broken v. straight is that since the broken has a 'nutcracker effect' it is generally harsher. By nutcracker I mean it folds together putting more pressure on the bars, tongue and roof of the mouth. Typically a solid kimberwick is considered gentler. I've only used a broken once, like I said I"m not a fan- I'd go with a solid kimberwick and set it on the curb ring (lower of the slots) if you have that great of a drive to go with a stronger bit.
And no matter what anyone says, never use a converter on a kimberwick! Those are only for pelhams!