This suggestion is getting me closer.
It turns out that you must have a 'use' statement that looks like this to get access to the $ua object:
use LWP::Simple qw/$ua/;
If you want to use get() without fully qualifying it, then you also have to include that along with $ua.
When I specify the '$ua' in the import list, everything starts working, and working well.
If I do not specify anything in the import list, then LWP::Simple::get() or just get() never returns.
It looks like my PPM is broken, too, so I think that I have a re-install in my near future. I must've gotten my hands on a bad module somewhere...