After switching to 10k resistors, the loaded battery voltage stopped sinking, and held steady at 5.28V. The Vo voltage measured in at 2.63V, which I would call pretty darn close enough. Thanks again!
What makes the voltage fall depending on whether I'm using 270s or 10ks? I went and did some reading on batteries just now, and it seems like the voltage should be more or less the same (this is an alkaline) regardless of load until the battery dies, at which point it decreases quickly. Is it because this particular battery is already down its death-curve quite a ways? In other words, if I use a brand-new 9V, will I see approximately equal loaded voltages regardless of what resistors I'm using?
These replies are really helping me reach an understanding of what's going on, and providing great pointers for further research. Thanks, lefty.
Irregardless, this 9V is toast, and it belongs in the battery recycler.