Yes, you want to shuck the action as you move the gun to the 2nd target...
Its a small point ....but pulling the trigger is the start of the shot, not the end of the shot ...follow thru is critical / and follow thru is the end of the shot. But the other thing - is you know there are 2 targets in the air .....but as you watch the pieces from the first bird disintegrate on the first shot follow-thru .....you want to shift your eyes to find the 2nd bird / not move the gun, until your eyes acquire the 2nd bird.
Play a game with yourself....sitting at your desk or whatever...
pretend you are executing a shot - say left to right ( just raise your trigger hand - and move accross the room - as you get to the crease in the wall / to the next wall - pull the trigger (on your hand ) and keep some follow thru on your arm..... Now, shift your eyes - to the left, where you would expect that 2nd target to be .... and then move your gun ....you'll find its easy to insert the gun in front of the target and track and kill it.... But try it the wrong way too..... move the gun (or your arm) without shifting your eyes first .... and most of us find, we overshoot the target, and kind of oscillate back and forth to establish our lead....vs being able to do it smoothly ....
the key, is shifting your eyes first, before you move the gun ....or you're all over the place. And once you have your eyes firmly fixed on the leading edge of that target ( not the butt ) ..always the leading edge ...it makes executing that shuck / and the 2nd shot a lot smoother and easier ...and you don't feel as rushed.
That's why I use the term "swat" at a target ...when someone doesn't shift their eyes first ...its just a prayer ...not a good move to the target. My Skeet buddy, says " count the rings on the target" ....and I always laugh, like any normal human being can really see rings on a target ( I can't ) ....but his point is "hard focus" ... and when I miss ...his favorit thing is .... "what did you see last " as you executed the shot ..... ( the barrel, nothing, don't know ....) none of which is good......
eye on the target, always the beak, never the butt... then shift your gun ...and I think you'll be fine... sounds easy doesn't it .... ??? / but it takes some practice ....
If you shift your eyes properly / then move and shuck the action / it'll all come together for you. If the action on your pump isn't buttery smooth / do what you can to smooth it up. My old pump guns ( both Browning BPS hunter models in 12ga and 20ga ) have thousands of shells thru them ....so they're very smooth. Some of the newer guns ..not so much ...so smoothing up the action will help a little.
You can practice this at home with snap caps too .../maybe with a mini-mag flashlight in the barrel ..../ lots of ways to get better with a pump.