For anyone who sees this in the future, I contacted Line 6 support. They had me reinstall the firmware, and that did not solve the issue. They sent me a shipping label and I've sent the Stomp to them for repairs. I'm due to receive it back in a day or so. I'll make an update then.
I did do some searching and have seen this issue mentioned elsewhere online. I think it's just random bad luck. Many times the problem is fixed by powering the Stomp off and on, but when that doesn't fix it, you'll likely need to send it back to Line 6. If it happens when you're outside of warranty, you'll be out of luck, I'm guessing.