I think I can relate to that a bit. I’m a trumpet player, but I experienced something similar to what I think you’re describing.

For me, I think it was more about the air–it is a very subtle thing, almost imperceptible at times, especially when playing music and trying to read music, get the fingerings, articulation, dynamics, etc. right and play with a group, etc.

The key me for maximizing the relaxation of the lips was the sensation of “leading with the air” and letting the lips respond to the air. It’s a split second thing, or has to be when playing certain music / articulations / pitches, but it’s important in ensuring that only the necessary tension is in the embouchure.

Also, thinking “Oooooh” as I’m playing, or “TeeOoooh” if articulating, even if my tongue is more arched in an “Eeee” syllable, helps me ensure I’m engaging the aperture corners and not the middle of the lips.

Another thing that helped me was someone recommended visualizing a coffee stirrer straw in their lips when playing higher notes.