radio does not shut off when key is out of ignition

Will it shut off when you open one of the doors? If so it is operating properly. If not possibly an ignition switch issue.

Check alternator output sounds crazy but a shorted diode or winding does crazy things

No, it stays on even when I open the doors. I can leave my car all night come back the next day (not turning the car on anytime in between) and I can turn on the radio without even putting the keys in the ignition.

Is it stock radio or an aftermarket? If aftermarket radio some people install them with a direct full time hot wire.

Have the opposite of that, i want to turn on my radio when ignition is off, but can't Do that, even if i turn the switch to on mode, if the ignition isnt fully switched on, i cant listen to music