GOOD NEWS (I hope)
I have replaced the require_once() function in all of my PHP pages, loaded to the GoDaddy server and they all now successfully interact with my MySQL database.
Rather than use the default Dreamweaver connection: <?php require_once('Connections/FOBA.php'); ?> I am using: <?php include("Connections/FOBA.php"); ?> which seems to resolve the issue I was having. Godaddy never mentioned updating their PHP to a version that deprecates the require_once() function. Any thoughts?