I have personal first-hand experience of this. The best advice I can give you is to continue reassuring your child as best you can, and under no circumstances ever threaten them that you will leave or that they will have to go live somewhere else if they don't behave - if you do that you are only feeding the anxiety and making it worse, it is literally the worst thing you can say, no matter how desperate you feel at the time. And believe me I understand, because with my DD it got so bad I actually self harmed on a few occasions, I felt like I was losing my mind.
The thing you need to get your head around first is that the behaviour your child is displaying is not coming from a place of naughtiness, it is coming from a place of anxiety, and while that anxiety still exists so will the behaviour. Punishments will not work, I can guarantee it - the anxiety will be more powerful than the fear of punishment.
My DD developed Separation Anxiety at bedtimes just before she turned 8 - it started on Christmas Eve in fact and I think it was triggered by anxiety about Santa coming in to her room, plus an overnight trip with Brownies that she had been on. She couldn't be left in her room alone to fall asleep, she would scream and cry if I left the room. For 4 months I had to sit in with her, sometimes for hours, until she fell asleep. We tried everything, literally everything - I could write a list a page long of everything we tried - reassurance, talking, discipline, help books - and nothing worked. We also had the problem that it was me, not my DH that she developed this clingyness to, so we couldn't even take it in turns to sit with her, it was all on me. I used to sit in the dark in her room doing the online shop on my phone just to feel like I wasn't wasting my evening. We gradually increased how far away from her bed I was sitting, until I was outside the door, but I had to sing nursery rhymes so she could hear I was there, and then it progressed to her knocking on the wall and I would knock back. It took about 7 months before I could go downstairs and I had to check on her every ten mins until she was asleep. It was a long and frustrating process. There were nights when she had to sleep in bed with me etc. It put a hell of a strain on our marriage as well, my DH threatened to leave etc, it was a nightmare.
Now two years on she still has separation anxiety and is nearly 10, but it is MUCH more manageable. She might sleep in bed with me once a week, it really depends. She will sometimes be clingy at bedtime but I am firm and about 60% of the time she will accept that and go to sleep. Other nights she is great and settles straight down. She reads in bed for 15 mins, at which point I pop back in and tuck her up etc. Sleepovers are totally out of the question, she can't cope with them at all. We have also noticed it gets better during term time, we think because she is in a routine etc. We talk through her anxieties, reassure her it isn't her fault and one day she will grow out of it.
I can tell you that on her bad nights she has said that she feels like once I am out of sight, even if I'm only in the next room, she is scared that I have disappeared and she'll never see me again. She also has dreams fairly regularly that me and her dad have died (usually eaten by a dinosaur or something!). We have gone through the school to try and get her some help, however it's been about 16 months since we made that call and we are still waiting for the initial assessment (apparently they are really busy!)
It must be really so much harder for you with a new baby, I can't imagine it. But your DH absolutely needs to step in and help too. Just remember, reassurance is key and don't make threats to leave or anything similar. Try not to let it escalate into a meltdown on your part, it won't help in the long run. I know that's easier to say than it is to do though. Good luck, I hope it's just a passing phase.