But I don't think the problem you are having is just that /etc/resolv.conf; I think that the system is actually using the new DNS server(s).
Let me see if I understand your situation: You are connected to a VPN via WiFi using DNS server A at your VPN provider, and you plug in the Ethernet cable, which, via DHCP, sets the DNS server to B which is not at your VPN provider. Then DNS queries are being directed to B, which is not what you want.
If you have "Monitor Network Settings" checked on the 'Configurations' panel of Tunnelblick's 'VPN Details' window, Tunnelblick will deal with this. It will restore the DNS settings and other settings changed by DHCP if they are changed to the pre-VPN values, or it will restart the connection if they are changed to something else. Some rare situations call for more fine-grained control than that; you can use the 'While Connected' tab of the 'Advanced' settings window to specify in more detail what should be done when a network change is detected.
I could be wrong about this, but as I remember, a "restart" of the connection may not be a full restart -- it depends on your OpenVPN configuration. I think it if you have "persist-tun" in the configuration, it is not fully restarted, but only partially restarted. That may be what is happening to you.