Ubuntu 16.04 DNS problems after 20.7.8_4-amd64 update

[jwketchum]

Shortly after final 20.7.8_4 update, my Ubuntu 16.04 workstation started having problems resolving domain names.  I have other Ubuntu 18.04 machines, windows machines, and android and ios machines, none of which seem to be having issues.  Attempting to look at the problem with Wireshark, I see messages of the type "standard query response 0xxxxx No such name [A or AAAA] [hostname]...." which are returned by OPNsense on DNS queries from the Ubuntu 16.04 machine, also returned by Google DNS servers 8.8.4.4 and 8.8.8.8.
Is this correct behavior?  It would seem that the flaw would be on the Ubuntu machine, since the rest of the network appears to be running properly, but I haven't changed any settings there.
If I reset the connection from the Ubuntu machine using Network Manager, DNS recovers but only for a very short time, and then stops working again.
Any debugging help would be much appreciated.