> power transformer taken from an old Altec consumer home stereo preamplifier which used... 12AX7s... it has two green leads (fairly thick) and on the other side two black, two red, and one yellow lead -- all equal but smaller gauge wire than the green.
First, be sure where it came from: 120-Volt-land or 230-Volt-land. In Plano TX, 120V is most likely, but beware gifts from Brits.
You know, from the 12AX7 use, that it is probably 120V, 6V or 12V, and about 200-300V. And that the 6V-12V is higher current than the rest, so the fat wires may be the low voltage. Or they could be well-insulated wall-voltage. But Black is almost always wall-power. And Red is almost always the Plate power (200-300V).
> I used my multimeter to check continuity of the windings, and only the green wired sounded.
Darn it, don't use a too-clever buzzer to measure transformers. The high voltage windings are higher resistance than a "continuity tester" generally reads, plus the high inductance really confuses many digital meters that pulse the test current. (I nearly threw-out a perfectly good plate transformer that way: digi-meter said "Open" but it was fine.) The old $15 analog meter is better.
> the black leads read 65 ohms while the reds measured 250 ohms (the greens measured 0.00).
Another clue that the Red-Red voltage is -probably- higher than the Black-Black voltage. And 12AX7s usually get fed higher than 120V power, if possible.
0.00 on the Green-Green is wrong. 0.1 is more likely. Again some digi-meters "cleverly" round-down low readings. I have one that doesn't, and it reads 0.1 ohm with the probes tightly shorted, which is why they force "low" readings to a nice neat "zero". The other possibility is that this iron is wandering loose in the world BECAUSE it has an internal short and was replaced, but didn't end up in the trash.
If you are brave, connect the Black-Black to 120V and see what comes out. If you are smarter than that, splice a 60 Watt lamp in series with the transformer and the wall: if the transformer is dead-shorted, you just get 60 watts of light, instead of blown fuses or fires. Be VERY CAREFUL: we know there should be hundreds of volts in there somewhere. Connect your meter with clip-leads with the power off, stand back, plug-in, wait for slow fire, then check your meter.
Or to be very conservative: get a 12 V AC transformer. Connect about 100 ohms in series with the 12V side. Connect that to the supposed 120V wires. Now all the voltages should be about 1/10th of what they will be when running 120V, and if anything does not make sense then you can think about it without any annoying smoke in your eyes or very deadly voltages within reach. If the Red-Red read say 20V, you have about 200V, suitable for about 270V DC. And the Green-Green should show either 0.6V or 1.2V, implying 6V or 12V heater supply. That assumes the 12V across the Black-Black stays 12V. If it is significantly less than the 12V feeding it, and that 100 ohm resistor gets hot, it is drawing unusual current for a small transformer and is probably shorted.