Well, I got the problem fixed.
I did a ton of reading all over the internet about people who have encountered the same problem, troubleshooting, etc. I couldn't find the source of the problem myself in my driveway, so I went to a local trailer dealer. The guy there hooked up a tester, and we were able to determine that the problem was on the vehicle side of the equation. He hooked up a new Prodigy brake controller, and it showed "NC" too, so the issue was in the wiring.
I went to an auto parts store and bought some electrical contact cleaner for corrosion and sprayed the 7 pin connector really good. No help. I then disconnected and took apart the 7 pin connector and found the culprit! Blue/green corrosion on the trailer brake connector and on the ground connector. This was inside
the connector, so no amount of work on the outside would have helped. With the spray cleaner and some scraping, I was able to clean it up really well, and the problem of the 'NC" code was no more!!
What I learned, though, is that this is no simple problem. It could have been a number of things. I'm very fortunate that my fix was a $5.00 bottle of spray cleaner.