Looks like it's fixed. For me, the process was:
1. Log into dashboard and find 10 new system updates waiting.
2. Take the updates and let them apply.
3. Turn the DNS module back on (that is, install it again).
4. All my local DNS entries were still there, but all the forwarders were gone. Entered a few preferred ones and tested.
5. Changed the DHCP config to use "local Zentyal DNS" again instead of "Custom: 192.168.0.1".
6. Tested lookups from workstations, all good.
Thanks for the rapid repair, devs!
R.