Hi OP
We have a new baby in the family, our DG - now a few months old. This is the second DG for our DS and DiL. Our DiL was telling us the other day that if the baby does not cry, etc for half an hour she gets up and looks and worries. She did the same for the first.
Our DiL, son who took a couple of months leave and even with our help, they were all very exhausted
IMO, and what I can recall of our babies over 30 years ago, it happens to most women
The next time the health vistor/nurse midwife come to see you, mention this to them for reassurance, help, and guidance.
Also speak to family, ideally mum/siblings if feasible!!
I'm guessing you will soon settle in to a routine. Also get the babies dad to do night-time duties on checking etc to give you rest. Day times, I guess you are both lacking sleep, have a little rota for naps etc