Flaky radio - 98 Taurus

I think my '98 Taurus is starting to feel it's age (120k miles). About once every three weeks or so, when I turn off the ignition, the radio continues to play for anywhere from 15 seconds to 5 minutes. This has been going on for at least three years - I didn't think too much about it, figuring it was a flaky switch somewhere, hardly worth the effort.
This morning when I started up the car, I tried to change the station, then the volume, then the function, then just turn it off, but nothing was recoginzed. I finally had to turn off the engine and start again to clear the problem.

The only other problem I have with the car is at night (at least that is the only time it is apparent.) I will be driving along on a flat road (so no need to increase RPMs), and the lights will dim for just a fraction of a second. It cycles through like that about every five minutes or so.

Does anyone else have a similar experience? Any guidance would be greatly appreciated.

The night thing is probably the engine fan kicking on.

My car does that with the radio also. I thought maybe it was just a loose wire. My lights at night do the dimming thing also but they do it a lot.