Ok, I got this problem tracked down, so I wanted to make one last post in case anyone else is searching on this matter in the future.

On the old machine, I had PHP 4/5. On the new machine, there was only PHP5. We changed this to be PHP 4/5 like before, and now everything works perfectly.

I have no idea why this script wouldn't function the same with PHP 5 alone. All I know is that making the switch cleared everything up immediately.