Ah i remember these days! It was similar for me: I'd be up all hours of the night trying to get my daughter to sleep in her Moses basket. Sometimes I would get say an hour-ish at 10pm, then not sleep again until 4am. Bouncing on my yoga ball trying to rock her to sleep. I used to listen to podcasts, watch documentaries, make myself coffee in the middle of the night to stay awake. Solidarity, I know how hard it is.
I never bed-shared. No shade to people that do, I was just too scared of squashing her or knocking her out if the bed! And breastfeeding was very painful for me so we wouldn't have managed that "barely waking to feed" situation.
Although the first two months were really, really difficult, she then was fine sleeping in her bassinet. I feel like it frontloaded the difficulty: things got much, much easier at about 8 weeks, from when we could put her to bed in her bassinet and enjoy the evening (or even have trusted people babysit).
In the short term, if you can express some milk, perhaps you can get someone else to take a shift with your son? My husband would sometimes take 5am - 10am, with her in the living room so I had the bedroom to myself to get some sleep.
This is the most sleep deprived bit, you will get more sleep soon !