My heart really goes out to you OP - those first few weeks are bloody shockingly awful hard work in my opinion. The sleep deprivation of looking after a newborn, especially an unsettled one, is just hideous. Take a deep breath: you are doing everything you can.
It might be silent reflux, which is like heartburn, and is bad when baby lies on it's back. Talk to your HV or GP if this is your instinct after reading about it: they can prescribe baby gavisconor somehting similar which can help.
I have to say though, my DS did not have reflux, but he was terribly unsettled in the night for the first few weeks of his life. He was basically nocturnal and would spend the whole night snuffling/crying/feeding and generally being a nightmare. DH and I used to tag team each other in and out of looking after him (ie he would do 8pm-midnight, I would do mignight till 4am, and then he would do 4am onwards. Even when he was at work, bless him. I was FF though after an epic breastfeeding FAIL so at least he could help out with the night feeds. Although tbh, DS was only feeding 2-3 times in a night, but was just awake, upset and unsettled the rest of the time.
We did not want to co-sleep so we battled on with the moses basket, basically none of us getting much sleep for the first few weeks, until he started to 'get it' and then became more settled.
The good news is: it passed. Just by him getting older, bigger more settled. I didn't really have to do anything apart form find ways to ride it out. I used to ask my MIL to come round for a couple of hours in the afternoon twice a week and I used to leave her with a bottle and just go to bed for a couple of hours. Could you express some milk and do similar with a friend or relative?
A wise friend told me it gets a bit easier at 6 weeks and then much easier at 12 weeks. I didn;t believe it was true (and thought I would have gone insane by 6 weeks anyway), but she was right. By 12 weeks he was sleeping for 3 hour stretches at a time in the night, and we were all feeling much better.
Hang in there, it wont be like this forever xxx