I know this is old but in case this helps anyone else, I though I’d post. It took me months to find the source of this issue in my 2003. Locate and pull the accessory power relay. It’s in the inside cabin fuse box. Your manual labels it ACC in the diagram. In a 2003 Escape, it’s right in the center of the fise panel. Use a multimeter, set to measure ohms, to look for a closed circuit on the non-energized relay crossing pins 3 & 5. You SHOULD see NO connection. IF you DO see a connection across 3 and 5, then your relay is stuck “closed” and this is why your radio stays on. Cheap, easy part to replace, but causes major headaches when it is bad.
$13 part at Autozone. Cheaper if you order online.