UPDATE
I fixed it! Thank you so much to everyone here who helped!
My findings:
It all hinges on the notches the spring slides in to.
If the notches aren’t lined up and/or the spring slips out of the notches when the brake is pulled, one of the brakes gets stuck.
So to center the brakes, focus on the notches not the brakes.
Whichever pad is sticking, make sure the spring on that side is resting on the bottom of the notch and not the top when you center the brakes.
Otherwise, when you use the brakes, the spring will get pulled down to the bottom of the notch and the brake on that side will get stuck.
My notch is much larger than the one in the image so there was a lot of play there.
