Ok, so I switched out the bit from the regular loose ring snaffle to the double jointed bit and that alone made a huge difference. She does seem to have a pretty shallow palate, her teeth seem fine to me, I'm no expert though. Nothing is swollen, cracked, no bothersom wolf teeth, she's not showing any other signs that her teeth are hurting her. The owner is planning on having her teeth floated in about 2 weeks though.
We were riding her on a loose rein before, which she's great with neck reining and responding to your seat to slow down, but I had my students ride with just light contact and had them hold fast if she stretched her neck down, not to pull back or jerk, but to hold her and squeeze if she bumped herself in the mouth and slowed down. She really didn't try it much and I think the bit had a lot to do with it, when she did try and my students closed that door, she picked her head back up and kept going. So, you guys were right about her evading the bit, and I think she was evading it so much because it hurt. I asked the owner if it was something she normally did and he said he honestly hasn't ridden her enough to know what was normal. He said he has ridden her probably 20 times in the 5 or so years he's had her. This man owns and operates an equestrian facility. A very nice one. Today went really great though. Thank you all for your help.