S.T.
both you and your husband should come up with a night routine where both of you compromise who sleeps and who tends the baby.
as for the baby crying, it's tough. each one has their own way to soothe and sometimes by the time you understand it, they have transitioned to another phase.
i have a 9 month old daughter. i remember when she was a new born, it was really hard for me to wake up and tend to her constant cries. but i learned that the sooner i would wake up to carry her and feed her, the faster she would go to sleep. and then that became our routine and eventually my body got used it. in my case, i had to since my partner was living out of state at the time.
good luck and have patience!! use your resources wisely