More testing.
The test was more complicated than normal because the trailer wires are not color-coded. All four wires are red, so I don't know what each wire does since I replaced the trailer plug and now removed it for testing.
- I disconnected the trailer's wiring and hitch from the truck.
- I used a 12v car battery to test and determine what each wire does. I connected the negative battery terminal to the trailer chassis. Then touched one trailer wire at a time to the positive battery terminal to determine what each wire does.
- First wire: Driver's signal light illuminated.
- Second wire: Passenger's signal light illuminated.
- Third wire: Sparks = Ground
- Fourth wire: Didn't illuminate any lights. Got very hot after a few seconds.
The fourth wire seems to be the running lights wire. I believed it was causing the issue and possibly shorting somewhere.
However, after thinking about it more, I figured I should check the light bulbs. I removed the lenses, and sure enough, the driver's bulb had one of two filaments burnt out. It looked like the burnt filament was touching the other filament. Maybe it was arcing, causing a short. Also, the passenger's bulb was loose and corroded. Although it didn't appear to be burnt out.
I replaced both bulbs: 1157 Sylvania Long Life Mini Bulb
Now, all lights work, including the running lights. Good.
In hindsight, I should have checked the bulbs as step #1; "try the easiest thing first". However, I had thought the fact that the signal lights were working meant the bulbs were fine. I wasn't aware of the concept of dual-filament bulbs at the time, so I had no reason to suspect the bulbs were faulty.
I did replace the trailer plug, which is good because it had visible damage. But I think the root cause of the problem was the bulbs, not the plug.
I also changed a burnt-out fuse on the truck (#25 in the under-hood fuse diagram "Trailer Park Lamps"; 2010 Silverado 1500 gas). But I think I burnt that fuse out during testing; it didn't cause the initial trailer wiring issue.