Katie, it's really hard to say for certain. I think that it is very common to have overwhelming feelings, especially at the beginning. I would say that it's also normal to have some fear and anxiety over something happening to the baby. I do recall with my first, freaking out when her cord stump fell off when she was just 3 days old and was bleeding. We called the on-call ped for that one! A couple weeks later she had a little bit of blood in her diaper and I totally flipped out and cried for a while. I held it together and went to the ped the next day instead of calling at night, but still I went to the ped for it.
As you ease into parenthood hopefully things will calm down a bit, but everything is new for a FTM and you often don't know what's normal and what's not and every little issue can feel monumental. The first time your baby bleeds is tough, even if it's a nicked fingernail or a bit lip (once teeth are involved).
All this being said, though, more extreme anxiety where you have difficulty functioning is not normal and should be monitored. I'm glad you have placenta pills and are aware of your feelings- this is definitely step one. But if things progress and you are really having a hard time, I would suggest seeking help before things get to a more acute level. My SIL had unresolved PPD (mixed with other issues) and it is still afflicting her almost 2 years later- no joke. She was literally just in the hospital and a recovery center over the last several weeks. Be aware that there can be a hormone crash around 2 weeks postpartum and then things tend to regulate a bit better after that if you're breastfeeding, especially (which I know you are). Also be aware that the sleep deprivation can really pile up and exacerbate issues. Since you're already noting that you are especially sensitive, it will be even more important for you to get help when you need it so that you can catch up on your sleep.
Keep talking about it and seeking support!