When a skulk shrieker is triggered, it doesn't move. There's not really much else to say. Both shriekers generated naturally and player placed shriekers don't work. They will make the screaming noises and summon the warden, but it doesn't move or emit the particles it's supposed to.
To reproduce you
1) Start a new Minecraft world (or use an existing one)
2) Place or find a sculk shrieker by either:
2a) Going into Creative mode and getting one
2b) Find the Deep Dark and find one
3) Make a noise to trigger the shrieker
4) Listen to it make the noise, but watch it not move
could you attach a video of it