Hi, glonmo,
Are your system date and time correct? We've been seeing some bizarre issues connected to computers where those have drifted--eithe because of a dying CMOS battery or because the system isn't always connected to the Internet to receive a time sync. That might conceivably relate to what this most resembles to me: a system that thinks it's way behind on its LiveUpdates--as these fairly often require a system restart to fully install (due to how closely security software must integrate with your Operating System). So if your date and time are correct, one other thing you might try is running manual LiveUpdates and rebooting, on each machine, until Norton tells you you now have the most current updates.
Failing that, I begin to worry about malware. So if none of the above resolve it, you might download and run the free version only of malwarebytes:
http://downloads.malwarebytes.org/mbam-download.php
Malwarebytes isn't as effective at keeping nasties off your system in the first place...but it does have a knack for still spotting them even after a system has been compromised by something new and sophisticated enough to deceive & disable any security software. Do not let it remove anything at this stage; just run it. If it finds anything, cut-and-paste the log it produces into a message you post here. That will help us assess whether we need to refer you to a specialized "removalist" site, where an expert with special training and tools can work one-on-one with you to pull even the worst malware out by the root(kit) haha
V/R,
--DistEd2