I have good news! It was indeed the 6080 tube! I got a couple cheap 6080 tubes off ebay. With both of these tubes, the crack is totally silent. And to confirm, I plugged the original tube back into the crack, and the hum returned.
Just for the record, here's another difference I noticed between the new tubes and the original: With the original, every time I bumped my desk (like with my chair), I could hear it through my headphones in the form of an echo-type sound (the crack is sitting on my desk). Even "little bumps", like just typing on my keyboard. On every key press, that was enough of a "bump" on my desk that I could hear that odd echo sound through my headphones. The new tubes don't do this, which further reinforces my claim that the tube was the problem.
Thank you everyone for your help. I really appreciate it!