So, an update on this:
I replaced all the plugs and wires. The 8th one looked like it had never been replaced. This didn't change anything, but I'm glad I did it.
Drove up to Birmingham, the stuttering kept getting worse and worse, and I noticed my gas mileage was awful just going down the road. When I put the truck into "tow mode" (shifted down a gear), the mileage improved. That was the last clue for me that the torque convertor was bad and the transmission was slipping.
Took the truck to the most honest transmission guy in Birmingham. He's a wizard. Drove it 5 miles and diagnosed it immediately. Said he could rebuild it, or I could "Go and get myself a cheap and easy junkyard tranny" but he didn't recommend that route. I told him "that doesn't mean what you think it means these days" :-D
Had him rebuild it. Slipping and stuttering was all gone. Shifts like a dream, very smooth, both the parts and labor are warrantied. But there was still a bit of hesitation when I hit the gas.
Tonight we replaced the Throttle Position Sensor. That was the last piece of this issue.
Truck runs like a dream now. Responsive and smooth.
You guys are all amazing. Thank you for all the help!