I don't have access to the wiring schematic, and I know that the first gen drl system was pretty complex. However, from your description you are checking voltage at the headlights with reference to ground? In other words, one probe in the headlight plug, the other grounded? If that's true, then it sounds like you are losing the ground to the headlights.
I would try this: as you face the rear of the headlight plug, the common terminal that carries positive power into the headlight should be on the left. To confirm this, unplug both headlights and probe either headlight plug while disconnected. You should have positive power at the left side common terminal.
Still with me? Once you have confirmed that the positive is on the left (I don't want you to blow a fuse, and I'm only 95% certain), then you hook the headlights back up, and supply a ground to either the center or the right side terminal at the back of one headlight. If it lights up, you've confirmed you are missing the ground. On most Toyota trucks before the 2nd gens, the ground runs through the dimmer (hi/low) switch on the column. I would suspect that your dimmer switch has failed.