Solution Found.
As I posed as a possibility in my original post - the problem had nothing to do with the perl script. It was a password problem on the NT Server. The 'root' user password was changed. But - where was the password even being referenced?
Answer: In the NT Services. There is a Service called Remote Shell Service. When the script is run on the Unix system, running a 'remsh' goes through the Remote Shell Service. The Service was started by the 'root' user. The Service does not stop, unless something makes it stop. It had been started with the 'old' 'root' password. I stopped it, restarted it with the 'new' 'root' password and the script on Unix runs perfectly. Again.
FunkyMonk & cdarke: I appreciate the input. But, as I mentioned in the (admittedly long-winded) writeup - this process has been working for 7 years. The code didn't change. Something outside the code changed, and I finally found that. And, it now works again.
Cheers!