After obsessively watching the few videos I've shot from the drone, I nailed down that a lot of the flight issues are coming from the props not all lighting off at the same time. There is a setting to control this, but it did little good.
I decided to dive deep into this part of the interweb and came across a forum post from mid-2014 describing my issue. The basic gist is that the firmware that ships on the KK2.1x is buggy and most folks flash a new ROM.
Diving deeper still, I found the tool to do it with (after installing Java 6, yuck) and a rather long video showing the nuts and bolts (I mainly needed to know how to plug in the USBasp I have for the board).
I have to recalibrate the board, but that's not a big deal. My new set of props arrives tomorrow, this time with many, many backups.
So, hopefully, we'll have the Ugly Duckling Too in stable flight in a day or so.