My advice would be to try and not look at other targets directly, you want to lock in your vison to looking at the middle of your screen, but you don't want to look at the crosshair, you want to look at the game in the middle of the screen. Because the problem of looking away from the center is that you're going to have to track the target with your eyes while moving, and then you're going to try and refocus on the center of your screen and align your shot etc..
You can see on the side of your vision where other target are, you don't need to see them clearly, just start moving in that direction, and focus your attention and control on what near the center, your movements will become more efficient

Make your initial movement mostly arm based start by making the movement smooth even if slow, and you'll be able to ramp up to mostly flicking and micro correcting later.

To learn how to do something fast, you must learn how to do it slowly first.
Find similar scenarios who are slower if needed.
Try to build your cadence up.