I haven't taken lessons in a number of years, but my teacher used to tell me that when I played notes higher on the register, my chin tends to move/push forward and that I need to keep it still. I thought about it today and am wondering if that has anything to do with my feeling that any high notes I play just aren't that pretty. She said it was a bad habit, but I can't seem to fix it! Anyone have any suggestions?

I'm sort of in that same state. For the first few years of playing bass clarinet I never had to go to high B or C. 2 years later I find myself playing them, and I always get tense playing them because I didn't even get used to them in the first 2 years of playing. Now I play clarinet as well, and I do find it a bit easier to play on the clarinet. All I can say is think of all the notes the same. Use your embroucher as if playing a G (no fingers) and keep blowing, playing all notes in all registers. You should see that you dont need to move your mouth at all, but some mouth movement is good to keep it in tune. As for me, if Im going to a high note I actually expand my mouth as if hitting a low note, and it seems to be working very well for myself. I hope some of this stuff helped. Message me if you need any more information ^_^