... When I measured its voltages, I got 5 volts on one of the plates. On the other side of the plate resistor, the supply voltage was around 250 volts, so something was seriously screwy for the tube to be dropping 245 volts across a 220k plate resistor. ...
You already know what's the problem.
Balloon-Rock Theory:
If you have a device like a triode with a plate load resistor, when the triode draws current the plate node will sit at some voltage between B+ and ground, because current through the triode causes a voltage drop across the plate load, and the resulting plate voltage is something less than B+.
If the triode is open circuit, no current flows through the triode, so no voltage flows through the plate load resistor, and it cannot drop any voltage. The plate node drifts upward like a balloon to the B+ voltage.
If the triode is a short circuit, it is as though the tube plate were connected to ground, and current through the triode is limited only by the plate load resistor (and cathode resistor, if present). The large current drops all or nearly all the supply voltage across the plate load, and the plate node drops like a rock to ground (0v).
You measured 5v at your triode's plate, so somehow it was drawing excessive current and dropped nearly all the supply voltage across the plate load.
... I tested the tube on my tube tester just to see what it would read. One triode read 120 (high good), while the other triode maxed out the needle past good into the red "? area". ...
Ultimately, the meter in ALL tube testers is a current meter.
It may measure current as a result of voltages applied to tube elements (emissions tester) or it may measure a current variation as a result of a step in grid voltage or applied grid signal (static- and dynamic-mutual conductance testers).
Each type measures how much current the tube passes under
the tester's operating conditions, and compares that current to an expected value for a good tube,
as denoted by the meter calibration.
So your tube passed more current than the expected "good value" on one triode. But you could know this would happen based on your in-circuit measurement of 5v at the plate (too much current).
Exactly why, well that's a different problem.
It's certainly not possible for one half of a dual triode to be gassy and the other half good. ...
Yes, if the problem is excess gas in the tube.
But one triode
could have cathode coating contaminating the grid of one triode and not the other; that might lead to grid emission, a loss of bias and excessive current. Or there could be some other fault in the tube such as a relatively low-resistance leakage path where there should be no connection.
Ultimately, There's some kind of contamination or partial short within the tube affecting one triode. Snce you can't get into the tube to fix it, the precise cause may be academic.