I tried manually charging the battery (not the smartest thing in the world to do... but i'm not always one to follow the rules) by using 2 D batteries and leads connected to the positive and negative of the battery.
After about an hour of sitting on the homemade charger, the phone booted up and ALMOST got to the main screen before it died. It even showed the orange charging LED for a few seconds.
My guess is that the battery is SO dead that it just won't hold any kind of charge anymore. If that is the case, it seems like crappy manufacturing to require a good battery to use the phone even when plugged in. It should at least attempt to charge it, or turn on when plugged in, etc.
I'll have to wait another week before my new battery shows up. Oh well.
Can anyone confirm that their 5600 phone won't power on when battery is removed but charger is plugged in?