Scott,
you are on the wrong way 
- You should start the Virus Scanner option.
- When the scanning process is ongoing, scroll down the Notification Drawer (I mean the area where Android shows all the notifications), and you will see Avast is scanning. Touch that line - it will pop up the “hidden” Avast screen where you can see the Avast Notifications. There will be a Clear button, you must use that, not the one of the standard Android Notification Drawer.
I still can confirm that this method is working for me. No old warnings came back since I gone through the steps Big Ron provided.
Good luck - if you still have problem, just post it! 