Generally,auto electronic accessories have more than one source of 12V DC power. This is a constant ON 12V source-so that things like station presets aren't lost everytime the unit is switched off.There is also a "switched" 12 volt source that may energize high current draw items like stand alone amplifier's or disc changers.

There is also a 12V tap going to the vehicles instrument lights-so that when the cars lights are on the radio indicators come on and can be dimmed with the rest of the dash.

Without a wiring diagram-all is speculation. Blown fuses are usually caused by wiring "mistakes". Damage could have been done to an adjacent circuit by accident. Trouble shoot your power sources from the unit back to the power source and I'll bet you'll find an OBVIOUS issue. Good Luck!