The removal utility of Norton/Symantec also leaves remnants. You may try the correct specific removal utility under Windows Safe Mode.
Then reboot and manually search for "norton" and "symantec" as being part of file names and folder names. The same for the registry. Delete them and reboot yet again.
Download the latest stable version of the same edition of avast you already have. Using "run as administrator", install that version over the previous one, and reboot after the installation finishes. Please avoid testing or changing anything else before the reboot.
After the last reboot, wait 10 minutes (with no testing) and reboot yet again (firewall tools may still have some "quirks" after the first reboot). Now, after the second reboot, wait again 10 minutes before testing the resulting behaviour (you won't need to wait always, just this time, for testing purposes).
Please report back.