Hi, sorry to hear you're going through this. I can relate, and I know how tiring it can be. I had three children I breastfed and was so tired at times, I was desperate. One of my babies slept through right away, one was average but one took forever.
I have a wonderful pediatrician very like Dr. Sears. Very no-nonsense, has a big brood himself, is very respected and down to earth. Here's the advice he gave me when I was desperate.
First, he told me that there are no hard and fast rules for when a baby should sleep through the night. He told me to forget about that :) That was kind of a relief, after having a first child who slept through super early, and I was following the same routine but it wasn't working.
He said the first step was to minimize the wakings, and that there were things I could do to prevent them in the first place.
He asked me if I was doing a really big feed before bed and during the night, or if the baby was falling asleep during feeds, or only feeding on one side. He referred those kinds of feeds as snacks, and said they will definitely wake sooner if they haven't had a full feed. So better to keep them slightly awake, and drain both breasts. Otherwise they will wake again in an hour or two.
He said let him cry just long enough to hear that "hunger" cry before feeding him. If we pop a soother in, then that might work for 1/2 hour, and you get back to sleep, only to be woken again.
He said often breastfeeding moms don't have quite enough milk at the end of a long day for the last feed or night feeds, so the babies will get hungry and wake more often. He said to up my water consumption especially towards the end of the day. He said some moms supplement with a bottle for the last feed, if they are really running low. Whatever works for the mom and baby.
Very important - he said keep the baby awake long enough to do a BIG burp. If you skimp on burps the last feed or during the night feeds, gas can become painful within an hour or so. He said better to wake them slightly to get it out, and have them sleep longer.
He said for some babies, being wet is enough to make them cry out so again, better to wake them slightly, and change them if this is the case.
For me, the best advice he gave was to buy a sleeper bag, especially if babies kick off covers. He said a lot of babies wake just from being cold.
He said do these preventative steps, and put him down when he's sleepy, and it should improve. It helped me, so I hope this was helpful :)