OP, I am so sorry. Having a miserable baby is way tougher than anyone who hasn;t had one understands. It attacks the bond between you, it attacks any shred of confidence you have as a mother, it isolates you form mothers with easier babies. It is a constant and profound source of anxiety.
DS2 was like this. Like your baby he was very ill. He was also autistic which we didn't realise and had a pronounced spinal problem which doctors kept ignoring - which caused him pain in pretty much all positions, especially lying down which is all babies can do at first.
They are in pain. They have no other means of communication. It is really hard on you and them.
My top advice is:
Become a tiger mother about his health. Just decide not to care that doctors think you are fussy, over anxious, attention-seeking. Grow a rhino hide and tunnel vision. Keep asking around. Our break through came when I literally accosted a random doctor in a hospital corridor and said, 'No one is listening to me, my baby is really unwell. I am sick of waiting. Come and look at him!' I literally dragged this poor man by the sleeve. By chance he was a specialist in two of the issues causing DS so much pain and he was superior to the pathetic doctor who was fobbing me off, so he insisted on new medication which helped DS get more comfortable.
My other advice is - if it drives you nuts and there's nothing you can do. take him out in a sling or buggy and walk down main roads with lorries roaring past to drown out the screams or around a park, wearing headphones (be prepared to get filthy looks) Or for a drive with loud music b;aring. not ideal but sometimes you really need a different noise drilling through your skull for your own sanity.
If you can, get help. I had none and I ended up with severe PND. Ask relatives to take him for a walk around the park so you can have an hour totally to yourself, in silence. If your sleep is very disturbed, pay for a night nurse and explain he is a sickly baby who cries a lot. If I had my time again, I'd take out a second mortgage to pay for a medical night nurse. Life is more bearable with sleep.
He outgrew it gradually, but it was a long, slow process. Now he is an adult, loving and so sweet natured, and very resilient.