After I turn the key completely off, the car takes a few seconds to "power down" (?!) before it's completely shut off. The battery light glows, the radio whines down, and the engine rumbles on a little. What's up?

If this is a fuel injected engine, the pumps should be stopping when you turn off the key. The engine can't diesel if the fuel flow stops. However, if any of the injectors are leaking then dieseling may occur as described by others. If the ignition and idle speed check out then I suggest a bottle of fuel system cleaner. Techron and BK seem to be favorites here.

I forgot to mention.......... The reason the radio and lights are still on while the car winds down is probably due to the alternator still spinning and producing current. This stops when the engine does and is probably not any thing to worry about.
Dennis

Several things cause "dieseling" most common is timing set wrong. On older cars another cause was build up of carbon in the combustion chamber but with the new fuels that is been pretty much eliminated. The timing being off can cause excessive heat in the combustion chamber like the other feedback says.This causes the cylinder to fire just like a diesel which has iginition by heat of compression. The older engines that had the build up of carbon would develop heat of combustion due to the compression being raised due to the chamber being smaller due to the build up. Way, Way back they used to take crushed walnut shells and with the engine runing would sprinkle the shells down the carb................. This would help break up the carbon and blow it out the exaust, then came diesel fuel sprinkled down the carb slowley until the car would stall. They would let it sit for awhile and then start the car, drive around looking like a mosqueto fogger for awhile until it all burned out. (the diesel fuel would help by cutting through the carbon like a solevent) Next was additives you could pour down the carb to clean it out. This was easy but had drawbacks. If you got too much down too fast you could "hydralyc" the engine as liquid does not compress and you had the danger of bending a connecting rod. I doubt you have any build up of carbon though because the newer gas does not leave deposits like the old stuff did.
Check your spark plugs and see if they are old, dirty or need replacing, check the timing and if carburated check to see if the mixture is set too lean ( this also causes the engine to run hotter. By hotter I mean in the combustion chamber, and it may not necessarly show on your temp gauge which shows overall engine temp.
check those items and also the idle speed ( not too fast) and I think you will have corrected your problem.
Good Luck,
Dennis

Sounds like you've got "dieseling", which I believe is caused by excess heat combusting fuel in the absence of a spark.