shelley,
Don't know for certain but as ZA is, by definition, monitoring your internet use, once you're offline it's dormant. If you shut the program down, I guess there may be some elements waiting for a startup command but they won't be actively checking security.
A real - time anti virus program should be active whenever the computer is on unless you tell it otherwise. With Norton that means disabling active scanning but I assume other parts of the program continue to function.
Disabling anti - virus is usually recommended when other scanners, like AdAware, are being used or if new software is being installed.
Have you tried Spybot?. It's worth a look and I know that you can run it and AdAware on the same machine because I do (but not at the same time!!!). Take a look at the 'Security Help Tools' list at the top of the Security forum here at TSG.