Yes, it really does get better.
The first bit is really, really hard.
Before I had my first DC I was at a posh wedding where two women were talking in the toilets about a third who was due to give birth. One said to the other 'Ithe thing is, it doesn't matter how much you want the baby, the first three months are always going to be utterly hellish, and the other agreed. But BOTH these women had maternity night nurses to do the night shift every night and never missed an hour of sleep!.....and they still thought that! It terrified me as I knew I didn't have the means for a maternity nurse!!
In retrospect, though, I'm glad I overheard it. As mainly all I'd heard was people saying things like 'you are in a little bubble of love' and not at all honest about it being hard.
Neither of mine would sleep if put down. At all. I found a sling very useful. Get up, change the baby, feed the baby, into the sling. Make my breakfast, get dressed, get baby's bag ready, leave house etc....get to where I wanted to go, baby out, change baby, feed baby, back in sling for me to eat etc....
I'd do random things like go to an art gallery, go to baby cinema, go to a baby singing group, yes I'm sure I looked a state and probably had twigs in my hair, but if I could get up, baby in sling, and get out of house, I could cope.
The danger for me was thinking i'm too tired, messy, fragile to go anywhere....that way is isolation and desperation....go out every day.