Well, I put the old pads back on and the issue is resolved. Aside from the actual thickness of the pad, there doesn't seem to be a size difference between the old and new pads. The new pads are Rotex Golds, so I assumed they would work. The actual braking surface of the pads was the same, just a difference in thickness.
I had put the teardrop-shaped shim on the new pads, and there also seem to be another metallic shim on the old pad covering most of the back, but wasn't easily removed, so I left it attached. Didn't do anything different when I re-mounted the old ones, so not sure what was up.