Ray Lamontagne is fantastic; good choice! I wouldn't leave music on every time you go somewhere; it might trigger the "something is different -- something is wrong -- I am scared" thought process for him. Just like with babies -- those motion swings or rides in the car will almost guarantee a sleepy baby, but you can't rely on them every single time just because it works... because what happens when you don't have the swing or can't take the car out? Baby freaks out because it is conditioned to sleep only when it is being moved in a consistent and soothing way.
You've only had Fawkes a little while; even though it is a big majority of his life so far, it's still going to take some getting used to with his surroundings and routines. My Maya absolutely HATED her crate. She would HOWL and CRY and WHINE and it just broke my heart. But guess what eventually happened? One day I walked into the room and door wide open, there she was, sleeping in her crate. No coaxing, no external motivation... she simply became comfortable with it being her safe and personal space. She absolutely loves her crate. We don't share crates at our house so it is her very own bed and her scent and her special place to be alone. But it did take a while for her to realize that the crate was not a solitary confinement chamber where she would never come out again. We picked the phrase "Do some crate time" for her, and we used it every time we put her in there. It's a conditioning response after a while -- we would say it as we picked her up and after we put her in, along with great praise "good girl!!!" So whenever she heard "Do some crate time", she knew when she went in she would get praised. It became a positive place for her. But remember, it did take some time.
And the trick that worked the best for us was a Kong filled with peanut butter. She would tire herself out and only whine for a bit. But it's important to not give in to the whining. As with enjoying being in the crate, going to sleep without whining will take time. It is a confusing thing for them to get used to, but routines really do help. Same time every night, same phrases, great praise... it'll get easier. Maybe even keep a weekly "journal" on what time you do things, what treats / toys you used in the crate, how much exercise he got [definitely makes a huge difference], etc. and you'll see trends in what works and what doesn't.
I'm looking forward to getting another corgi pup in a couple months; then I'll be in your shoes with the sleepless nights. ;-) Good luck!!
I would turn the music off at night when you turn off the lights. Night is the natural time for the dog to sleep, just as it is for us. I would leave it on when you go to work. Anything that soothes him is welcome, as it is very hard for a pup to be alone for any length of time. Dogs are pack animals, not loners.
I agree with Anna, music during the day and not at night. Have you tried a kong with cheese or something smeared inside? It would give him something soothing to do. I only give this treat when I am leaving so that makes it more special.