Yeah. Everything is easier if stay still.
My suggestion is isolation of skill.
Staying down on pool stroke is a skill.
Try first shoot shots directly to pocket without object ball and just focus staying still. Shoot fast and slow speed shots. You will notice it will come easy very fast.
Then just spread all 15 balls and shoot them in with cueball. Idea is to get used staying still while more game like situation.
Dont care anything else than staying down. Missing or position dont matter. Success = staying down
You will notice how staying down while aiming cuts is a lot harder than without object ball. It is natural, because some of our focus goes into that.
I made video to my little sister about this subject years ago when she did have problems with staying down.
I do example half hour practice. You can notice i stay better still on my shots than normally because im focusing it.
I recommend doing this routine 2 weeks daily and you will improve faster than you could imagine.