Tips for getting sleep (newborn trenches)

I am EP my three week old son. Sometimes I nurse him but he doesn't latch well at all so I just comfort nurse in the morning when he wakes and before bed for a few minutes after he gets his bottle.

He is currently sleeping just around two hours, usually less during the night. My husband and I do shifts but I still get up to pump as I don't want to lose my supply. I sometimes skip the 2-3am pump because I am so tired. The problem is with baby waking right around every 90 minutes to eat at night and then having to pump every three hours, I am struggling to get sleep. Even when I try to take naps during the day or when my husband has him from 8-12, I just can't seem to fall asleep. I am walking around like a zombie...until I lay down. Any tips or tricks? I've tried white noise, meditation, taking a warm shower...nothing seems to help. It takes me like an hour to settle down enough to sleep. It's really hard right now. Any advice would be appreciated.