Appreciate all the advice so far, but things aren't looking too good here.

I started by pulling out every single fuse in the driver side dash fuse panel (after taking a picture so I knew how to put them all back again). Tested and still lots of battery drain.

So I put all those back and tried removing different "blocks" of fuses from the panel under the hood.

Got most of the way through that, still seeing the same results on my meter, until I realized if I held the probes down for more than a split second, the really high reading (pegging the meter) stopped and it dropped back down to something more measurable.

(Now I realized that initial really high reading could have just been whatever relay clicks when the car firsts gets battery power, and is normal.)

I really need to get a better, digital voltmeter and try all of this over again. This ancient analog meter doesn't even make it clear to me what I'm seeing on the readout. (I know I want to set it to "DCA" mode to measure current draw, but the only two settings on the dial are "50u" and "250m", and the scale on the dial doesn't even have a place where they're labeling milliamps of current or anything. (They have a green part of the scale for ohms, so maybe I'm supposed to be using that scale and doing some sort of conversion?)