Tube Troubleshooting

No, tube dampers would not help that problem, and it does not sound like a microphonic tube (or tubes). It would be unusual for both tubes to go bad at the same time though. Before disposing of the "bad" tubes, I would first have them checked. What brand of 6SN7 were they?

Yes, at least one of your 6SN7's are bad. Hopefully you live close to someone who has a tube tester, and they can check them for you. I don't have a tech or someone with a tube tester close, so I am looking at getting one for myself.
Thankfully, you found it right away. Cheers -Don

This sounds like a classic case of tube rush. Tube rush is typically found in older spent tubes. The confusing element here is that the OP spoke of more than one tube. It is very unusual for 2 or more tubes to start rushing simultaneously. Thus the suggestion to have them tested is good one.
You asked about microphony, and whether tube dampers would help? The answer is no this is not what microphony refers to and yes tube dampers may help with true microphony.
Microphony is a sensitivity too feedback sounds/ noise or percussion from the environment. Microphony is usually tested by keeping the suspect tube in circuit with the volume at audible but not loud levels, and tapping the suspect tube. If the tapping noise comes through the circuit to the speakers ( plays out the tapping,) you have microphony. It is almost very common in the earliest 6SN7s meaning the 6SN7GT. You don't have dispose of microphonic tube IMO because you shouldn't be tapping or drumming on your tubes while listening. Tubes that are sensitive to sound pressure feedback, are no good and need to be disposed of. (you could try dampers see below, but in general they are too far gone.)
Dampers can help with microphony but have no effect with tube rush which is a symptom of old age and cannot be reversed or helped. In some heavily microphonic tubes using anti vibratory dampers can help, but you don't hear most microphonuc tubes in thre first place.