1. Both speakers or one of the two?
2. I doubt this is purely a grounding problem, although the hum problem might be separately related to grounding. Again, is the hum in both speakers or only one? And is it really "hum", defined as 60Hz or 120Hz tone, or is it "buzz", which sounds higher in pitch and is really a mixture of frequencies?
3. My thought on the fact that sounds made when you handle your tonearm are transduced into the speakers is perhaps a microphonic tube. Most likely culprit is the first tube (input) of the phono section. That's typically a 12AX7, and it supplies much of the gain of your phono stage. Check that there is good acoustic isolation between your tt and your phono stage or preamplifier.
Like someone else said, I would also say that if the problem is in one channel, then you can check hypotheses by swapping tubes from one side to the other.