While it seemed like unchecking "Enable HTTPS scanning" corrected the problem, I did still have problems with websites not loading up on occasion, but it was a lot less often than before.
I tried unchecking "Enable QUIC/HTTP3", but that did not seem to correct the issue... however, websites loaded up much faster with that off.
I tried a different DNS with everything enabled in Avast!, and it had the same connection issues as the Cloudflare DNS... plus, it was a lot slower.
I tried enabling everything again in Avast!, getting rid of the DNS settings, switching back to the ISP's DNS, and that seemed to correct the issue. I still need to test this some more, but there is something triggering this issue on multiple computers.
Is there something else that needs to be done to use different DNS settings in Windows 7? I was changing the DNS through the network adapter settings.