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The heater is 0.55 volts at 0.06A or 60mA. That is 0.03 watts.
You would need a 100 ohm resistor in series with the CRT filament to run it from a 6.3 VAC transformer. The resistor would put out 1/3 watt. Nothing should be getting warm at all.
Did you put a resistor in parallel with the filament instead of in series?
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