I use livewell stoppers…the black ones with the little lever. There are different sizes and they are somewhat adjustable for the different sized holes.
You are correct: unscrew the screen and put a livewell stopper in it. I do mine from the outside because I can’t reach down in there all that well.
I plug ANY hole that could possibly take in water and then freeze. I have 6 holes out the back and two holes near the front for livewell overflow.
I do not use my livewell at all during cold weather fishing. I know that some people put fish in the livewell “dry”. Not me…p-youey! Bring a cooler if you want to keep fish.
Hope that helps!