I have this too. I have a history of anxiety. If you read 'What Mothers Do' it actually says about this. The author suggests that it's more about a warning of what could happen. For weeks I'd wake up simply relieved she'd made it through the night massive fear of SIDS DD is 5 months now. These thoughts still pop into my head but not so severely. I think sleep deprivation plays a big part. I felt my brain was saying You're really tired, mind you don't drop her on the kitchen floor! If it doesn't improve, a trip to the GP is in order though.
Yes, I had this for a few months - worse with DC1 than 2. I found it exhausting - felt if I was ever less than 100% vigilant something awful would happen. It did pass, and as others have said, there is advice out there. actually with DC2, where I had anxiety due to other things happening at that time, I took anti-depressants and things turned round v quickly. I think feeling dreadful is made even worse because you have expected to feel so happy, and you feel guilty for not enjoying your baby. I hope you feel better very soon.
Google intrusive thoughts. This is what you are describing. It is part and parcel of anxiety. It may get better or you may need help. There are good strategies online for dealing with them. You are not going mad!
I think your baby is tiny and to feel like you don't want to be away from them, or in a different room, is completely normal. I felt like this. Secretly I didn't really even want anyone else to even hold my baby for the first 6/8 weeks! Your hormones are all over the place at this stage and I think it is natural to want to protect your baby.
If you are unable to enjoy your DS or sleep even when you are with him because you are worrying so much, I would go and have a chat with your GP, but otherwise sounds like part of being a new mummy. Congratulations on your DS. I found weeks 3-4 the hardest, so hopefully will calm down for you soon.