I just did a full rear (drum) brake change in my Cobalt. I bought new drums, painted them, and replaced the shoes and hardware. (not including the adjusters) Everything went great until I drove it and I get a very noticeable vibration when stopping from over 30 mph or so. The frequency of the vibration is proportional to my speed. I feel it mostly in the brake pedal but the whole car shakes. Suspecting the drums to be the issue, I took them back to O'Reilly's where I bought them and had them turned on their lathe. The drums spun true. Their next advice was to do the stop in reverse a few times. I did that and it maybe helped a little but the problem is definitely still there. I'm not an expert on what the lathe actually does, but is it possible that there is something wrong with the drums that wouldn't show up during the test?

Also, I took everything apart to check it and I noticed the drums stick a little in the same spot on every rotation. I think this is the cause of the problem but I don't know the root of it.

I feel like I've done all the troubleshooting I can think of. Any ideas?