It’s freezing here, but I did finally have some time to get the truck up on jack stands and run. Instead of noticing wobble, I found the rear left wheel not running whatsoever–I still need to check the differential type (and I admittedly know very little about this), but I believe this is still suspect since there’s no traction issue or slipping going on, correct? Could my issue be having 1WD?
This is not abnormal for a regular differential on jack stands. A differential is designed to allow one side to rotate at a different speed from the other to enable turns, which causes the wheel on the outside curve to have to travel a longer arc length in the same timeframe. When on stands, if there’s more resistance on one side the side with the least resistance can spin all by itself. As a matter of fact, if you (with the engine off) manually spin one wheel, the wheel on the other side will turn in the opposite direction. It all has to do with the way the pinion and ring gears, the “spider gears”, the carrier housing, and the axles all interact. The pinion on the driveshaft turns the crown gear, the crown gear turns the carrier housing, that pulls two of the spider gears to orbit the spider gears on the axles in a planetary way, and can result in the carrier housing, the pinion and crown gears, the orbiting spider gears, and only one axle to operate while the other axle and its gear remain static.
For the record, I’m having a hard time describing this in words. Any help would be appreciated. 
Which raises another question. Has anyone looked at your differential?
As TSM posted above, that’s normal for a non-locking differential. It wouldn’t do that on my truck b/c it has a limited slip differential. I presume both rear wheels are off the ground for your test. Did you have the wheel installed on the left but no wheel on the right, just the hub? If so that could be the reason the left wheel isn’t rotating. There’s more inertia needed to get the wheeled side moving , so only the non-wheel-sides rotates. I think if you remove the wheel on the left side (or install the wheel on the right) and re-do the test both will probably spin. If the brake adjustment differs from side to side, that could cause this effect too. If so, back off the adjusters so both sides freely spin for the test.
When I was in high school I had an embarrassing incident involving this effect. I was on a first date and got my RWD car stuck on a busy street right in downtown Denver when one of the wheels somehow got suspended in the air above a deep gutter … lol …
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You couldn’t find a more remote spot to get stuck in? 
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I actually tested this idea of resistance by doing the following:
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Mounting the tire on only the right side–right side still moved, left remained unmoving.
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While the truck was idling in drive, I carefully applied heavy resistance to the right hub with a wide piece of particleboard–no change on the left side.
To answer your question, the differential has not been checked. I’m not sure what type of differential comes standard on an '03 Ranger RWD, and I can’t seem to find it anywhere as it’s not listed on the replacement OEM parts.
Normally we would remove the brake drums and measure the run-out of the axle flanges. If you don’t have a dial indicator you can set up a pointer like a large screw driver resting on a jack stand pointed at the axle flange where the drum was mounted. Turn the axle by hand while observing the distance between the tip of the screw driver and the axle flange.
Based on your observations about the left drum being stuck, if the left axle is bent the drum will wabble causing the brake to drag. Rotate that drum by hand to see if there is obvious irregular drag.
Thanks for the tip. Should I have the drum secured with lug nuts while I do this?
Remove the drums and examine the axles.