The flapping noise on the bottom of the door was really driving me crazy. I was worried I needed to rip the entire jamb out to fix it, but I decided to start with a more subtle approach. Turns out the door just wasn't sitting quite flush in the frame, causing the latch to hit the strike plate unevenly.
I checked the hinge mounting screws first. Two of them were slightly recessed and loose. Tightening those gave me a bit of movement, but it wasn't enugh to stop the wobble. I then focused on the clearance at the threshold, realizing the strike plate needed to be shimmed slightly outward so the door could swing home and lock without binding. It took some careful measurement and shimming of about a half an inch, but once that adjustment was made, the door slid into its proper groove. The whole thing stopped bouncing. It was way easier than I expected.