NM 1.2.4: Problem with OpenVPN DNS lookups after Ubuntu 16.10 upgrade

[Paul Smith <psmith gnu org>]

Hi all.  I recently upgraded to Ubuntu 16.10 and have been having
intermittent issues with DNS lookups ever since.  Unfortunately the
infrastructure on my system has surpassed my ability to debug it: gone
are the days where the DNS servers simply sat in /etc/resolv.conf, or
else in simple DHCP lease files.

My system is a desktop system, on a LAN, but I use openvpn to connect to
work and I also have kvm/qemu installed for running virtual machines,
which does its own playing with resolv.conf and its own dnsmasq in order
to enable virtual LAN facilities.

network-manager                   1.2.4-0ubuntu1
network-manager-gnome             1.2.4-0ubuntu2
network-manager-openvpn           1.2.6-2ubuntu1
network-manager-openvpn-gnome     1.2.6-2ubuntu1
network-manager-pptp              1.2.2-1
network-manager-pptp-gnome        1.2.2-1

I've configured my system to use Google DNS servers by default and
normal access to the internet works fine.  The problem is that when I
start my VPN, I can't look up any hosts on that network's internal LAN. 
The local DNS servers are not getting queried.  The VPN is working fine:
if I edit my /etc/hosts with hostnames I'm interested in then I can
reach them just fine.

My IPv4 settings for the VPN are all automatic DHCP, DNS, Routes, and I
do check the "Use this connection only for resources on its network".

When I first upgraded to Ubuntu 16.10 and rebooted I had this problem,
but then I rebooted again and it went away so I figured it was an
upgrade glitch.  But now it's back and I'd like to get to the bottom of
it.

Can anyone tell me how to investigate / debug this issue?  My
/etc/resolv.conf has:

  # Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8)
  #     DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN
  nameserver 127.0.1.1

so clearly something is taking over DNS.  I expect it's this dnsmasq:

  /usr/sbin/dnsmasq --no-resolv --keep-in-foreground --no-hosts \
      --bind-interfaces --pid-file=/var/run/NetworkManager/dnsmasq.pid \
      --listen-address=127.0.1.1 --cache-size=0 --conf-file=/dev/null \
      --proxy-dnssec --enable-dbus=org.freedesktop.NetworkManager.dnsmasq \
      --conf-dir=/etc/NetworkManager/dnsmasq.d

but I've looked in those directories and I can't find anything that
looks like it might be a DHCP lease file or whatever that might tell the
system what DNS servers to use (in fact /etc/NetworkManager/dnsmasq.d is
empty)

Help!