I had similar problem in my 2002 A6. My turn signal lights would come on, but not blink. I popped the fuse out for the blinkers and put it back in. Turn signals started blinking again. Similar fix for your radio lights staying on after removing the key. Find the the fuse, pull it out, reset it, and everything back to normal.

 

I had a vehicle do this once, turned out the previous owner many many years before had used the point of a test light to pierce the wire to test its voltage, water got in it and corroded from the inside. I found the pin hole, opened it up (the wire) and sure enough it was done.