Unfortunately to say, training is more important than anything else here. Your dunsparce seems to have grown used to hearing the radio, so he’s becoming distressed when it’s not on, as that represents a drastic change in environment. So what you’ll need to do is wean him off the radio using a bit of replacement therapy.
Start off with small time increments—five minutes, for example. During those five minutes, switch the radio off and immediately engage in some other activity, such as training or playing. (Of the two, I would recommend beginning with playing, as you may be able to use toys to distract dunsparce enough to keep his mind off the radio. The toys can then be used to keep him distracted while you’re not at home.) If he becomes too distraught to listen to you, switch the radio back on and try again later. If, however, he follows along with the activity, reward him with a treat. Keep doing this until he’s comfortable with having the radio off for the period of time you’ve set, then increase the time more and more until he can spend all of the hours he’s with you away from the radio. Finish off this training by reinforcing that from the time you come home to the time you leave the house, the radio will be off and that the time you’re not at home is essentially “Radio Time.”
That said, yes, you can still play the radio while you’re not at home, but so long as your dunsparce understands the difference between what is and isn’t “Radio Time” (through the training you give him), he should be easier to handle when you need a bit of peace and quiet at home.
Best of luck!
The Pokédex According to Bill 