Honestly, you're not the only one. I am the same. This is my first baby, but I'm not sure I'd be any better with my second if I had one.
Soon after my daughter was born, she developed a bit of nappy rash from pooping constantly. I was convinced she had a serious form of flesh-eating dermatitis, until I started using Sudocrem and it cleared up almost instantly. While jiggling her gently, at four weeks, she was a little bit sick and I became terrified I had caused shaken baby syndrome! Before she was three months old, I had already paid to have her examined by a private paediatrician for an incredibly rare eye cancer I was convinced she had. When she developed breastfeeding problems, I did too much Googling and convinced myself she had cerebral palsy and couldn't swallow properly. I'm still worrying about that now, even though she's not missed a single milestone and is getting ready to walk any day now. I watch her constantly for 'signs' that essentially aren't even really there. Googling has been my downfall.
I don't know why I'm like this, but I think it has to do with being aware that there is so much that can go/be wrong. It happens to other people, so why not to me? The only plus is that if there really WAS a problem, I know I would notice it straight away.
My anxiety is extreme, and I hope yours won't become as bad. Just rest assured I often feel like I am at my wits' end from all the worry, so you're not alone!
I say definitely get a monitor, so that SIDS is one less thing to worry about!