I had written “I think electrons are moving into different suborbitals without transitioning between shells”.
I figured that the different m_J levels represented different arrangements of electrons in orbitals and suborbitals and their spins. Since the spins can’t change, and a change in orbital would mean a larger energy transition, I figured the only thing that could change would be the suborbital they are occupying.

I don’t think this is really off-base. I still think this is the only way the J vector could be changing orientation, as was the final explanation.