It will settle down, I promise you, although it may take a while. It's very early days, too, so don't put pressure on yourself or your DS - nobody magically adapts to a new addition to the family without some stress and upheaval.
My DS was 3 yrs 9 mths when our DD was born and he was a nightmare for a good few months. Stroppy, disobedient, extra clingy/needy, extremely jealous of his DD and would physically attack her etc. It was very hard on us all.
Get the school on side. Go and talk to his teacher and explain that he is finding it hard to adjust and ask for any suggestions of ways you can all support him through this transition in his life. he is only little and it's important the school understand whats going on and help him through it.
Evenings - can you and your DH split duties? One of you puts DS to bed and makes it a really nice, 'special' time with stories and hugs etc, while the other deals with the baby. Then swap around another night?
No matter how hard it is, make some 'special time' for him each day. Even if it's just 15 mins. Build it up and make a big deal of it. I say to my DS all day 'We're going to read stories / do drawings / make cookies tonight, just you and me'. Really hype it! . Do it at weekends, too. I starts leaving my DD with DH for an hour on Sat mornings and taking DS for a hot choclate at the local cafe or to the library or for a walk in the park, just us. It was a break for me, too.
Don't be hard on yourself, either. It's a massive adjustment for you, too. You need time to recover and get your strength up, physically and mentally, and you need time to get used to the changes in your life.