As a mom who did full time care of my 2 as hubby couldn't afford to take any time off, I know your frustration. I'm proud to say my son did at least 50-50 care with his wife and when she did some overtime on Saturdays, he took the baby and often would take her for the day just to give his wife a break.
But he developed one sneaky trick I wish I had thought of when he was little. Babies cry instead of talk and when are tired, they just let it all hang out. So he'd get her all changed and made sure she was comfy in her car seat(they used it inside too) and then he'd put her in a darkened bathroom and turn on the ceiling fan that takes the moisture out of the bathroom and close the door most of the way and she'd be out like a light in no time. The quiet whosshing if the fan, the low light of a single night light and she would sleep for a good 2 hours plus. Try it when he gets really fussy.
Remember. he just came from 9 months of listening to the constant beat of a heart, blood rushing by, digestive noises, darkness and warmth......re-create that same situation and he should calm down and sleep.
It takes time to get used to moving into a new neighborhood and for you it takes time getting used to living on less sleep. But you'll all adjust.
At least you didn't have the situation one of my friends just had. They waited a little too long to head to the hospital and baby was delivered by a cop on the side of a 4 lane road, in the car, in the dark. Now that's the way to start an new life!