Thanks for all your replies on this issue. I really appreciate everyone's help.
Currently, Avast is working normally. Not sure if I fixed it, but here's what happened.
I recently added a dual-band USB Wi-fi adapter to my laptop whose built-in Wi-fi was 2.4GHz only (thus limiting my download speed). This addition achieved the desired speed boost from around 60Mbps on 2.4GHz to around 180Mbps on the 5.0GHz band of the USB adapter. So far, so good. What I didn't notice right away is that, when I woke the laptop from Sleep later on, the new adapter wasn't reconnecting to the router. I wasn't trying to use the internet right away so I didn't notice that I had no internet.
This is when I think that Avast was trying to check for updates or otherwise "phone home", and it couldn't dial out, so to speak (does anyone remember dial-up internet @ 56Kbps?). Anyway, an Avast window would open with blank grey background and the message "Loading ... this won't take long", and the message didn't go away.
Eventually, I noticed the lack of internet connection when waking the laptop from Sleep. This led me to check the adapter's Properties window / Power Management tab where I unchecked the box next to the option "Allow the computer to turn off this device to save power". Now, after about 24 hours, the internet connection works when I wake the laptop from Sleep, and that Avast "Loading ..." window and the attendant red exclamation mark on the Avast system tray icon have not recurred.
Another option might be to adjust the settings in Control panel / Power options so that the laptop go-to-Sleep option is set to "Never". I don't like that option as much, but it would probably have the same effect for keeping Avast + internet working.