she holds her head up annoyingly high. She's in no frame at all
If she is holding her head up, then it means she is not relaxing her back and stretching back into a "frame" (I hate that term almost as much as headset!) Her back is hollow and she needs to learn to use her hind end. Have someone check your saddle, it could be pinching her somewhere making it uncomfortable for her to stretch down.. This could also be affecting her lead issues.
If your saddle turns out to fit fine, then try lunging her with side reins, start them out loose the first couple times so she can get used to it, and then start shortening them (the inside rein should be 3 holes shorter). Make sure she can still move her head, the side reins should NEVER be so short she cant stretch down or put her head up if she wants.
Lunge her for about 15 min, walk/trot, and then only ride her for about 20, walk trot. Until she is balanced and using herself properly.
If I touch her with my heel to push her over to one side, she just goes faster
Thats because your heel is the cue to go faster, instead of using the heel to move her over, use your calf, and say over until she understands. (like you would on the ground)
I would suggest just keeping her at walk and trot for now, cantering will come easier and better when she is going consistent and the walk and trot.
Good luck and I hope this helps, if you have any questions or something needs clearing up please feel free to PM me.