usually a radio stuck on is a sign of a bad ignition switch.
However, glad you provided more details i.e w/ button no reaction and only off when engine off.
So... my recommendation is to first pull down the faceplate and clean the contacts on faceplate and mates on radio either w/ rubbing alcohol or a pencil eraser.
Otherwise, your power button may be going bad or have something jammed up in it (like the peeling button coating
). In which case, take some compressed air (i.e. like the stuff used to clean PCs and sensitive equip) or toothpick / paperclip (even an artist razor) and clean out around the button b/w it and the radio chassis.