One week later, and I have made HUGE progress!!
I finally found the hum loop in this gadget! It was staring me in the face the whole time. I knew I had to be dealing with a ground loop that was somehow only rearing its ugly head when both "halves" of the circuit were running. Each stage by itself was clean. So, what was happening only when both were in the works? After going back and re-reading everything I had about grounding, I kept being reminded of a very basic (novice-level basic, at that...) rule. A ground loop is caused when there is more than one path to ground in a circuit.
So I sat there staring at this thing, looking at where there could be two paths to ground for the audio circuit. There it was, right in front of me the whole time. Right next to the power supply, I have a terminal strip where the various supply rails come out. Makes it easier to change things during the design phase. Anyway, two terminals are for the B+ supply and the 0V ground. There were two wires going to 0V. One came from the board for the first two tube stages. The other came from the PP stage. All this time I had treated them as two separate circuits with an interstage transformer between them, almost as some sort of "demilitarized zone."
I guess I also figured that, ultimately, they both ended up at 0V anyway. I kept thinking of it as a star ground. I forget which document I read that got me thinking in terms of a bus ground (think ground bar on a point-to-point rig...). So, I walked my way from one end to the other, and instead of connecting the PP ground to the 0V terminal, I connected it to the same ground run as the rest of the stages.
The big, nasty hum disappeared.
I could still hear a slight buzz, so I set about shuffling a few things. Discovered that by rotating the supply transformers a little I could get rid of that. All that was left was some hiss. Believe it or not, actually replacing the first tube with an OLD one got rid of that!
Added phantom power, and the noise stayed away.
Added an instrument input, and the noise stayed away, until....
When I plugged in an instrument, it started buzzing again. Tried a few tweaks to that input in the way of grounding, shielding, etc., but just can't seem to get rid of the buzz when something is plugged into the instrument input. Not a deal-breaker. I was happy to not bother with that feature anyway.