Hey op, you have my sympathy....sleep deprivation is a nightmare.
I have a succes story for you: my DS was a hideous sleeper around the 3-4 month mark. He was bottle fed and we gave him a large dreamfeed (7oz bottle) at 10pm when we went to bed. He would often then wake every hour or so in the night cos his flippin dummy fell out and he couldn;t soothe himself back off to sleep without it. a few times I tired feeding him but he pushed the bottle away. He wasn't hungry. Just awake and couldnt re-settle. One night when he was 3.5 months old I was in there every 45 mins putting it in and I thought this has to stop!
Me and DH decided to start to teach encourage him to learn to fall back to sleep himself when he woke after a sleep cycle finished in the night, without needing his dummy to get back off.
We started by tackling the early evening period first (ie 7pm - 10pm) cos we were more prepared to deal with stuff/crying/issues at that time of the night than at 4am, for example!
So we picked a day. On that day he went to bed around 7pm as usual. Then he started crying at 8.15pm as usual, asfter 1 sleep cycle had finished and he didnt have his dummy to go back off. We looked at the clock and sat on our hands. We said we would leave it 4 mins and see what happened. 3.5 mins later, he went silent. He had gone back to sleep! I never thought it would happen. But it did. previously i would have rushed in there to shhh pat him and put his dummy back in. He just needed to be left alone to figure it our himself.
Happened again at aroun 9.30pm that night. That time he cried for 4 mins and didnt stop so I went in and put my hand on his tummy and said shhhhh shhhhh. and then went out. He then grizzled/fussed for about 1 minute more then went back to sleep.
For the first few nights we didnt do this 'technique' in the wee small hours of the night....we just wanted to guarantee he would go back to sleep asap so we jumped right out of bed and gave him his dummy. But in the early evening we did the same thing for a few nights till he got the hang of it. Then we decided to see what happened in the middle of the night. By this stage he was nearly 4mo. I remember the first night we tried it: he woke around 2am and cried. I looked at the clock on my phone and decided to lie there and watch the clock for 4 mins (seemed kinder to him than 5), and lo and behold, 3.5 mins elapsed and then he suddenly went quiet.
Since this time, he has been mostly able to soothe himself back off to sleep in the night. There have been times when he hasn't gone back to sleep after 4 mins (my arbitrary 'line') and then I have always gone to him and tried to soothe with just a shhhhhh and a hand on his tummy or back. if this doesnt work I have picked him up for a cuddle. This has usually only happened when he has been coming down with a cold or cutting a tooth. A couple of times he has ended up in our bed usually before getting a cold the next day. But mostly he can self soothe (now 11mo).
I think there's a lot of anti-sleep-training younger babies info out there, but I think there's a big difference between leaving a small baby to wail for a long time, and sitting on your hands for a few mins to see if he'she can just drift off again on their own. I mean, I never would have known if DS could do it, if I didn;t try?? And in a way I think it's the kindest thing I could have done: he learned an important new skill: that he can go back to sleep on his own in the night. But it he's upset or in pain mummy or daddy will always come to him cos he wouldnt go back to sleep if he was in pain, right?
not sure if you're BFing or not, and if so, things might be a bit different, but I hope you find this story heartening :)