Blue,
Norton and Avast cannot be on the same computer as they both run resident and will conflict. Even though you turned off the Norton Resident scanner, it is still configured to scan your emails.
Avast will not "overpower" Norton and take control.
What may happen is that neither AV will scan the mail neither knowing which is default.
I suggest that you make a choice between them. If you want to remove Norton, it must be done correctly as Norton embedes itself deeply into the system.
You can obtain instructions on its' removal at
www.symantec.comThen we can go over your configuration of Avast.
By the way....What is a blonde day?
Take care and welcome to the forum.