So I managed to get my brakes sorted today!
I took the pads out of the front caliper and carefully squeezed the lever a few times. One piston was extending and retracting properly but the other was barely moving. I lubricated it with mineral oil and then squeezed the levers and pushed the pads back in a bunch of times. It started working a lot better after that.
On the rear brake, I noticed that the pads weren't seated firmly against the pistons, and I could wiggle them a bit. I tried replacing the retraction spring with the one from my road bike's Shimano 105 brakes (factory assembled and working well) and that held the pads in place much more firmly. I'm using some cheap Aliexpress pads and I guess the springs they come with aren't very good. The Shimano springs have a couple extra bends so they hold the pads against the pistons more securely. Luckily I had some extra Shimano springs and they got these pads fine, so I replaced both the front and rear.
After these changes, I was able to center the calipers by eye and the rotors only needed a few small tweaks to silence them.
I didn't have time to test ride it (and won't for a while) but it worked great on the stand!
