When time stops it stops in a reference frame. There is no motion in that reference frame but the frame can still move through spacetime relative to other reference frames. Think of light as a permanently frozen object. It pops into existence and does not change. It can still move through spacetime. The concept of instantaneousness does not apply to frozen objects. They don't experience time ever.

Similar to motion, a reference frame where time stopped can spin. That's how the singularity in a black hole behaves. Time is stopped in it but it still spins. When matter falls into the black hole angular moment adds up so it can spin up or spin down while still experiencing no time (remaining frozen).

Note that light frequency is not a movement in addition to the movement of light along geodesic lines. A frozen object can have internal frequency, spin, and momentum.