New poster:
Here's a partial goofball theory that's worth checking out:
I am coming in on this conv late, so if I've missed something, pls forgive. If the screen grid, nominally connected to the cathode either internally in the tube or externally via components or, it should be noted, via the UL winding taps were somehow NOT connected, then its (the screen grid's) influence would not be there.
This tube (not really this "tube", this primary branch of the OT circuit) is acting and measuring like it is running too hot. We would scarcely believe that a tube would lack this internal "jumper" from the factory, and you say the problem lives in the socket, not the tube. So I would tend to think that whatever cooked the socket (that made you have to replace it) might have overheated and partially fried something in the OT. Or, could it be that you have an amp that was started on the way from being populated w/6L6's and was intended to be made into a EL34 amp, the completion never occurred, and then someone put it away for ten years, looked at the tube chart, and threw 6L6's in there??
OR, an empty pin is being used as a tie point for the screen resistors and there is a hidden short you're overlooking..?
Goofy, I know, but cheaper to fry a few brain cells thinking about it than either tubes or trannys.
Keep us informed!