i noticed when i was practicing lip trills that there were two main ways to change the note. with my air, and with my jaw. the air approach gives me more control but i can do the trills faster with my jaw. what's the ideal way to do this? is it better to use the air or the mouth or a mixture?

As long as you have both notes clearly distinguished, and not an extendet vibrato as a till, both ways are fine. The result is what matters.

To really get fast on your trills I do not think you will be able to change the air speed between the notes and the jaw movement is a little too much.

What you can do is take the combination and minimize them. Move your jaw movement to only tongue movement and your air speed to air direction inside the mouth.

In a perfect world your lip trill works like this . . . . .
By using the vowel shapes of "ah", and "ee" you change the direction and speed of the air stream. Resulting in the two different pitches. There should be minimal jaw movement and you should be able to get a pretty quick trill going.