That would explain why everything was fine when I had 3 cards, but broke when I added the 4th.
Meanwhile, I'm still trying to figure out how to properly use OverDrivenTool. I'll get there... eventually.
EDIT - well, threw caution to the wind and just started changing stuff and seeing what worked and I made some progress. Got the new core and mem clocks to stick on every card and set a target temp of 64C for the fans. Saved the profile and next I'll see about automatically applying that profile either at boot or right before starting a miner (pretty much only Cryptonight or Ethash will be run on this rig).
EDIT 2 - and some weirdness, of course: looks like Cryptonight hashrate is up on the 1st and 4th cards, but not the 2nd and 3rd. GPU-Z reports the new clock rates for all 4 of the RX 560s, though, and I used the same 2-thread/intensity config for all cards in xmr-stak.