It is often the case that the DB structure changes with updates so I recommend backing up the db before upgrading. Support can help with getting this running again.
As per the direction of Support, I had to uninstall 3.6, restore the entire DB (not just the NTA parts), and run the config wizard for 3.5 again. Disappointing to lose the data from the few days while I was waiting to learn how to downgrade. I suppose that will teach me for doing an update directly to our production install instead of on the bench first.
Thanks again Andy.