It has become clear that I need to practice my transfer techniques; I just completed a primary-to-secondary transfer which involved what I'm guessing is a fairly significant amount of aeration. During my transfer (1-gallon basic brown beer, direct from carboy to carboy; no intermediate pot), I used an auto-siphon, and ended up pumping oxygen in. In addition, there was air in the line even while the beer was flowing (which I had to restart several times).
Ultimately, if a beer gets aerated, is it a solid bet that the beer will be oxidized and pick up that "musty, cardboard, stale odor and flavor"? Should I not get my hopes up, or might things still work out?
Last night, I opened one of the bottles. Much to my surprise, it was not awful! The carbonation was a little low, it could have had more flavor, and it was a little stale, but it was not the disaster that I was expecting. Hooray!