Transformers have no moving parts but they do have alternating magnetic fields due to the alternating AC current and so they can vibrate. Its usually small and silent when the alternations are balanced but even a tiny little bit of DC offset can make enough of a difference to cause it to vibrate to where we can hear it. A lot of DC offset and you need a DC offset eliminator. Or it can be so tiny the tricks oldhvymec suggested will work. Cleaning contacts and all the rest are all great things to do anyway.
As much fun as it would be to tell you yes this is a solid state thing you need a tube amp sorry, not true. You do need a tube amp but for another reason called sound quality.