I'm sorry you're going through this! It's hard to function well with a poor sleep pattern, and having a newborn just compounds it!
Sleeping is a function of rhythm, habit, comfort, and allowing yourself to let go. Nightmares and night terrors have many causes, but one to check that really works woth our little boy (who wakes up screaming) is that he has absolutely nothing sweet at least two hours before bed. If he does, he wakes up screaming. It's a simple equation!
My ohter suggestion is something called sleep therapy.
Basically, you go talk to your child in a soothing voice right after they fall asleep...go in about 15 minutes after he's asleep and just start talking to him. Tell him positive, present-tense statements about being safe, sleeping all night long, being comfortable...also include positive statements for his self-esteem, stuff about how much you and Daddy love him, how special he is, how glad you are that he's yours. Really tell him a lot about how he's safe and it's okay to sleep deeply. Getting this message into his "subconscious" mind, where the arousal has been programmed in, is a very powerful tool for correcting a pattern.
Good luck with that!