@NickTransmissions
I got the the valve body removed today. Here's what I found:
1. The roll pin and retainer for the manual low control valve had come loose, and the spring was hanging slightly out of the bore. I found both the pin and the retainer in the transmission pan (parts 21 and 22 in the TH350C ATSG book on the valve body exploded diagram). When I found this, I was hoping this was a smoking gun for why it wouldn't shift to 3rd.
2. When I got the valve body on the bench, it had some small rust particles visible. This was a used valve body I got on eBay after the original was unusable, and I knew it was rusty when I got it, but I thought I cleaned it extremely well. I guess I missed something.
Despite the bit of dirt in the valve body, all valves still moved freely, including the 2-3 shift valve. Since I didn't like the rust in the valve body and I had since bought another, much nicer TH350C valve body used on eBay, I decided to go back together with that one.
The valve body I installed today was spotless when I got it. I went ahead and disassembled everything just to make sure it was clean and moved freely. I also installed the shift kit springs (TransGo SK-350) into this new valve body.
I went for a drive, and it still won't shift into 3rd.
One thing I noticed, and I'm just grasping at straws now: Reverse doesn't feel right. If I don't physically hold the shift lever up almost to the park position, the truck will go into reverse but seems like it slips. If I hold the shift lever as close to park as I can get it, the truck goes into reverse firmly and doesn't slip at all when moving backwards. It did this with the old valve body I took out as well as the new one I installed today.
Also, when I got the truck all of the lockup stuff was still hooked up. It still is. I'm not sure if that could somehow keep it from shifting, but I'm 99% sure the truck did shift to 3rd before I rebuilt the transmission, and all of the lockup stuff was plugged in back then too.
Also also, and this may completely be my imagination: I've run the truck up to 55-65mph in 2nd, and it seems like I hear a slight whine from somewhere, almost like how a power steering pump sounds when it's low on fluid. I'm not sure it's coming from the transmission, but wanted to mention it in case it helped with the diagnosis.
Thank you again for your help.