Poking around in some books on nmr today I found someone who directly addresses the doublet of doublets phenomenon. You can't imagine how happy I was to see this.
I'm beginning to get it. As usual, Straumanis is right. The book explains that in most situations, nearest neighbor hydrogens are similar enough to one another that the splitting occurs in a way that follows the rules for splitting we have learned. In a few situations, though, when the neighbors are very different from one another (as expressed by something called their coupling constant), they split the signal sort of separately and generate a doublet of doublets.
The book provides an example that looks a lot like the fluorinated benze we were examining, except it has a nitro (NO2) group across from the F.
The fluorine is the reason for the different coupling constant in our molecule, but I am still sorting out the details of what role it plays. I also still don't have complete clarity about why these coupling constants could vary when the coupling has all been explained to me using the quantized spin states (as I tried to describe in my earlier post).