So, I figured out how to fix my situation. I re-installed my new cable and took another look at how it was actuating the detent valve in the valve body of the transmission. As I mentioned earlier, when I adjusted it by the book, the valve still had about 3/16" travel left at full throttle, when the valve really needed to be close to fully depressed, and that's how it still was. So, I tried an experiment. Instead of opening the throttle lever on the carb wide open before locking the snap-lock button on the cable for adjustment, as is the factory procedure, I unlocked the cable and rotated the throttle lever only about 3/4 wide open before locking the snap-lock button the cable. Then I opened the throttle to wide open and checked the valve again to see how that affected the valve travel. Now it was fully depressed! This in effect made the cable adjustment shorter, so when it goes to full throttle, it pulls the cable a little further than before and the kick-down valve becomes fully depressed. I also checked it to make sure the valve was being depressed sufficiently by using the gas pedal, and it was (thanks "tayto"). So, I test drove it and it shifted fine and I have a reliable kick-down function now. I just drove it on a 40 mile round trip to a friend's house this evening with no problem. I still don't know why it's not adjusting correctly by the factory method, but I'm glad I found a fairly easy way to make it work. It takes a little trial and error adjusting it this way, but it only took me 2 tries. Now that I know what's supposed to happen, in the future, I'm going check the detent valve actuation in any th350's I use before putting them in use.
Thank you both for responding!