Thanks for your reply, and the additional background on IPv6, DNS, & sysadmin stuff! 
Hardware: Librem 14v1
This occurs with both the Wi-Fi & Ethernet interface
I haven’t manually changed any of the default configuration that PureOS shipped wiith. For IPv4 this is set to Automatic (DHCP) by default on all networks, and IPv6 to Automatic. Both have DNS set to automatic. I’m not doing anything fancy like static IP or custom DNS servers. I imagine with the home routers I’m connecting to, these should all typically be a IPv4.
For the IP address, this configured correctly and gives me a 192.168.x.x IPv4 address depending on the router’s preferences. In GNOME settings, it does list the correct DNS servers from the router like 192.168.x.1, though I guess somehow it is not reaching /etc/resolv.conf.
Currently connecting to Google’s just as a troubleshooting measure, don’t want that to be a permanent fixture. It’s clear that something else related to the Network Manager is writing to /etc/resolv.conf, though if I knew that would likely help solve the issue.
I checked in /etc/NetworkManager/NetworkManager.conf, and it has the following, which doesn’t look unusual.
[main]
plugins=ifupdown,keyfile
[ifupdown]
managed=false