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If you have a detachable face unit, you can just turn the unit off by detaching the face and your settings (station pre-sets, clock) should remain intact. You can keep the PSU on.Over 30 years ago, I hooked up the radio (AM/FM analog stereo radio only) from my Mom's '77 Ford Granada to a 12 volt supple and the only problem I had was that the light stayed on after I turned the radio off. I had to unplug the 12 volt supply to turn the light off, Also...shouldn't you have an antenna attached to the unit? I did for mine.
Also check Crutchfield. They have wiring harnesses for almost every make and model.