Once upon a time, back in the days of the original XBox, I ran into a vaguely similar problem with one of my games and it turned out to be the result of cached game data being corrupted. According to some tech specs I found, the 5 (give or take) most recently played games kept a chunk of their data cached on the reserved portion of the hard drive in order to speed up initial game loading (and it made a huge difference with some games). You could tell you hadn't played a game in a while when it suddenly seemed to take a lot longer to load up the first time.
Armed with this knowledge (and an NHL 2K6 the suddenly got stuck on the Loading screen when trying to exit a completed game, both franchise and quick games) I ran though all of my other 360 games in order to try clearing the cache (I also manually deleted all NHL 2K6 saves from the System Blade of the Dashboard). I don't know what the magic number happens to be yet, but after running over a dozen other games I returned to NHL 2K6, fired up a quick game, and successfully exited afterward!!!
I'm now going to start up another franchise, wait till I can't exit again, then try repeating this process without deleting any data and running fewer other games inbetween (I'll start with 6). Though this would still be a significant inconvenience (assuming it works), at least it would make my seasons playable again.
If, however, my saved season remains broken, then I'll have to try it again with injuries turned off and make sure I never explicitly edit my lines ;). I'll report back with the results of all my tests (probably in a few days).
-Lone Jedi