Some good suggestions here. I’m going to suggest a different path. I see two potential culprits:
1) tube microphonics. It’s possible your tubes are introducing distortion from resonance. Only way to really tell is to try something different, which, of course, changes the sonic character overall. But better tube quality does not guarantee lack of microphonic excitation.
2) Get ready for the eye rolling from the usual suspects in these forums, but I banished my last bit of harshness with the addition of the Reiki Audio superswitch and linear power supply. I had a different upgraded (modest) in there and I was skeptical if it brought anything to the table but kept it in anyway. Marginal improvement at best, and perhaps my imagination. The Reiki brought immediately apparent improvement. I’m sure there are other switches that would do the same, but I can personally attest to this one. Digital artifacts are hard to banish and none sound good like 2nd order harmonics do from tubes. At least worth an audition.
My $.02.
Good luck.