Aw I really sympathise. I went through the exact same thing though carried on breastfeeding. He cried and screamed ALL day from 8 days old unless he was feeding or sleeping. He got drastically better around the 2.5 months mark (though the past day and half has been awful - hopefully only temporary... I think he has trapped wind. He's 3.5 months now).
So we initiqlly thoight he had trapped wind and found infacol and gripe water did nothing. We went to the local health clinic and paediatrician a LOT! Eventually we discovered he had a tongue tie, torticollis (neck injury) from ventouse birth, and suspected reflux, possibly from an allergy (still unsure).
The tongue tie was snipped and he started physio for tge Torticollis. We sae a cranial osteopath 3 or 4 times but he screamed the place down and i honestly thing it didbt help at all. He was put on ranitadine for the reflux. Trouble was, by then he had been so used to screaming all day that he hadn't been napping (I guess he was too uncomfortable) and all our efforts were yo get him to stop screaming rather than to get him to sleep (interestingly he's always been an amazing sleeper at night though thank god). So by 2.5 months, despite being on medication he was still really unhappy - and I now think at that point it was because he was overtired from no napping. So we starting following a vague version of The Little Ones programme, with three main naps a day- I think 9am, 12pm and 4:30am, or whatever works for your baby. The key thing is to not let them be awake for more than 2ish hours at a time. And nights are supposed to be 7-7. We started actively trying to get him to nap, introduced a dummy, tried the shush pat technique... and he was a MUCH happier baby a few days later! However we could never have done while he was in so much pain and discomfort. But it showed me how there are often multiple interconnected issues.
I now think he has lactose overload as I think he has high supply. He's an amazing happy chap most of the time now at 3.5 months but does go through the occasional spell when I think he's in pain and very windy - and today he's been worse than he's been for more than 5 weeks! However hopefully it's temporary.
I would say be really assertive with the doctors and ask to see a paediatrician. You could see your local health clinic at your children's centre too? They are the ones who referred him to a physio. Babies should not scream constantly all day long no matter what they tell you. And in the meantime do whatever you have to do to survive. It brings tears to my eyes to read your message as I remember so clearly the hell it was for me. And it's the not knowing if/when it will get better that's so hard!! Get your daughter checked for a tongue tie too if you haven't already - that could explain the long feeds. My son was checked in hospital as he couldn't latch but they said he was fine. It took a month to get him diagnosed at a baby cafe And by then we were stuck on nipple shields.
Best of luck and feel free to vent on here - those first two months of screaming were the hardest of my life and I was genuinely getting depressed- but it does get better o promise. And the main thing is, don't blame yourself. I was constantly comparing my baby to my NCT friends' happy, easy, contended babies but that was unfair to myself and my son! X