Is it a Samsung model by chance? I had one that did that a few years ago. When I asked customer service, the answer was that the phone jack doubles as an antenna. No laughing from me, the issue bugged me enough to call customer service. I thought that it was my fault, like I had hit a wrong button that messed up my phone.
I had to find out what it was we were not supposed to laugh about, but after reading your question, I couldn't help the laugh that escaped from me because I remember wondering about the same thing some time back, only I was too shy to ask anyone because I didn't want to look dumb...
There is no question that is dumb unless you already know the answer to the question. I had the same problem as I am also using a Samsung phone. What is funny is that I can play music through the player without the headphone but I cannot use the radio without the headphone connected.
While this has never happened to me, I'm laughing because it's exactly the type of thing that would! Sometimes these gadgets have the simplest explanations or fixes. It's almost too obvious to consider until someone tells you. Glad you got it all worked out, happy listening.