It’s healthy to feel the weight of what you are doing.
What I mean by that is this... you are aware and sensitive to the gravity of the situation you put yourself in when you fly an airplane. That is good! You are manifesting that knowledge and sense of gravity as negative emotion, that’s not ideal, but you can use it your advantage.
Use this feeling of dread to keep you vigilant, so that a dreadful thing does not happen. Or better yet, use the feeling to spur you on to Readiness, so that when the dreadful thing does happen (which it certainly will, as you have experienced) you will be prepared.
Use the feeling as your cue to always be on the lookout for emergency landing fields.
Use it as motivation to perform thorough preflight EVERY time.
Use it as motivation to track time and fuel burn. Use it as motivation to talk to ATC and pick up flight following. Picking up flight following will go a long ways toward easing your mind and feeling less alone.
The listening to music under your headset idea is not the correct antidote for what you are experiencing, I feel. I think you need to use the feeling as motivation to “sharpen yourself” so to speak, with vigilance and proactiveness, not to dull yourself by covering the feeling up with music.
That’s not to say music in The cockpit has no place... I just think it is the correct antidote to your problem. Your mind is trying to tell you something. Will you listen, or try to drown it out?