Mark, I had exactly the same issues. Trust me, the cable is definately bad!
"And why did it go back-and-forth okay when I put it in neutral?" When you put it in neutral (button depressed), the shifter actually does not move the gear select cable at all, it only moves the throttle cable. Yep, there are two seperate cables on the control - throttle and gears.
When mine was dying, I found that if I worked on it I could get it to work OK for the day, but the next time or the time after that, it was jammed again. Go for the Teleflex Extreme cable and your problems will be solved. Not expensive, and took about an hour to fit for someone inexperienced like me
For someone who does this for a living, would tak 15-20 minutes I expect.