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Resistance measurements are almost never any good to fault isolate a problem in a power transformer, unless it is an open circuit in a winding.
In the case of a short, usually what happens is adjacent turns in one of the layers in a windings short together. This only reduces the overall resistance of that winding by a fraction of an ohm, which is undetectable. But the effect on the transformer is dramatic. All the power in the transformer is delivered into that one shorted turn. That shorted turn overheats and causes the symptoms you describe. But a short EXTERNAL to the transformer will cause much the same results. You need to unhook ALL the secondary windings of the transformer. For the B+ circuits, this can be accomplished by simply removing the rectifier tube(s). For the filaments, you would need to unsolder at leas one of the wires on each of the filament secondaries.
Once you have done this, apply 115VAC power to the primary, and if the transformer still groans and gets hot, it has an internal short.
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John Folsom
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