It's an old problem: NetworkManager overwrites /etc/resolv.conf with each connection, forcing one to use the local ISP's crappy DNS servers instead of one's tried-and-true public DNS servers.
The fixes for this are legion: chattr resolv.conf, disable dhclient, replace NetworkManager.
It turns out that dhclient can be configured to include specified DNS servers (up to twp before a warning is displayed) in the resolv.conf.
Simply add the following line to /etc/dhcp/dhclient.conf :
prepend domain-name-servers 18.104.22.168, 22.214.171.124;
Obviously, replace 126.96.36.199 and 188.8.131.52 with whatever name servers are desired. The semicolon, by the way, is extremely important -- don't leave it out.