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The reading on pins 3 and 11 should be an open circuit. The path from those pins to ground goes through a zener diode. The meter will read it as an open circuit.
Seems to me they ran a jumper wire under each socket between pins 3 and 11.
I would check for continuity from pin 3 on the tube that gets hot to pin 3 on the tube that doesn't.
Likeliest thing to make the tube overheat would be loss of the negative bias voltage on pins 3 and 11 of one tube, but not on the other.
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