Thanks for the help, Heimdall!
I did as you suggested, but the broken games are still exhibiting the same behavior. I tested on three different games to be sure. Two of the them freeze on the splashscreen, and the other just freezes the Xbox and kills all input to the monitor--which I know because it automatically switches to another active input (my computer). This is the same behavior as before I deleted the cache and the contents of X, Y, and Z.
Any other ideas? Thanks again!