When I cut my headlights on my 05 Dodge Magnum the radio goes off

So the radio has a direct connection with the headlight switch. That is why the radio lights come on the buttons when you turn the headlights on. The connection at the radio or inside the radio is suspect. You could easily remove the illumination wire from the radio, and I bet the problem would go away, but there are a few reasons this might not be the best. The wire is hot any time the headlights are on, so you have to worry about it grounding to something. The radio may be a pass-through for the dash lights, meaning the illumination circuit lights the radio, then the radio sends powers other dash lights. And the radio buttons will not be visible at night. I would start by removing the wire to see if the problem resolves. Then reinstall it to see if the problem returns. Then I would either repair the wiring harness, replace the radio, or disconnect/bypass the radio on that circuit.