M.J.
Just hang in there.
It takes a lot longer than a month for things to reach a new "normal." My daughter is turning 2 next month and I can count how many evenings of peace we've had on one hand.
It's hard. It's stressful and yes...it's the exhaustion that is taking it's toll...not the baby. She is doing what babies do best...rely on you both.
My husband and I understand that we are to put each other first before our child because without a foundation on which to stand, the whole house will fall. Now with a baby in the mix, it's finding the balance of putting each other first while having this little person who can't be put second in line because she needs just about everything. If the two of you find a way to work together on meeting her needs, it will help you too to stay focused on knowing this is only temporary. She will be big soo fast....and you will wonder where those sleepless nights went.
My husband made an upspoken deal with each other, when the other was REALLY tired, the opposite would catch a second wind and try to carry the load. Allowing each other time to rest in shifts helps makes the transition easier. Keep putting each other first above your own exhaustion. Keep verballing affirming him, "what a great dad you are" so he doesn't feel just used and abused. I'm sure you feel the same way but you start the ball rolling and see what happens. With a baby, you can work and do and clean and care for and all you get in return is crying and fussing. It's nice to have a reward for your efforts. You be the one to give it to him and I'm sure it will follow for you.
Also, find a sitter and have date nights. You both desire some time off...together, guilt free.
Good luck. It does get better..still hard and you're still tired but there is more joy once she starts talking... it's the "Ya you" and the kisses...oh the rare sweet spontaneous kisses...that melt your heart and refill you all over again!