Hello, I never post on here normally but I just wanted to reassure you, I have been exactly where you are, twice, with both our babies.
The post by Douliket is so true, she sounds great, her advice is really sound. Advice for 'normal' insommnia isn't always that relevant - I've had that too, had CBT for it and therapy, and a lot of the advice isn't relevant really here because you can't try the usual things, because you are caring for a baby. The cause is also different because in this case your brain is quite literally programmed by pregnancy and birth to be super alert, to listen out for threats to your baby. I believe it is perfectly normal albeit horrifically unpleasant.
I thought I wouldn't have this crippling sleep related anxiety with my second, now aged 11 weeks, but I did, but because of receiving some helpful advice like that from Doulkilet and also from having been there once already and survived, I knew it was only a phase. Even now almost 3 months in I get it some nights, maybe once a week, even though my baby and toddler are good sleepers, I lie awake on the edge of panic, sometimes all night. It is utterly grim. BUT I know it won't last, even though it feels like it, and I know I will be fine, even though it doesn't feel like it. And you will be fine too. I think 3 weeks was the worst stage for me, with an almost 3 year old toddler too.
When I've had a terrible night, if I get a chance the following day I lie down to rest in the bedroom with the curtains drawn for 20 minutes if I can, whatever time it is, I don't even try to sleep. And even just 20 minutes lying down really helps.
I also use white noise. I have a little white noise machine because I hate knowing what time it is at night. I keep it low so I can still hear baby and toddler but I listen to it and somehow it helps. And my DH goes to the toddler if need be, even it means he screams and takes longer to settle, most of the time it is fine if I don't go.
Good luck.