I should have continued troubleshooting before posting, it appears there are all sorts of internal networking issues too.
Packaged Contact Center Enterprise 12.6
Boxes can't talk to one another [consistently, at least], web UIs for various tools won't open because there's [apparently] nothing listening at the far end, etc...
When things DO seem to be responding well you then get things like "This Finesse Server is currently out of service. For more information, contact your administrator."
In that instance, I powered off the Finesse server and brought it back up, with no change.