Tiny new babies look so vulnerable, don't they? I worried like anything about my three GC when they were new, terrified about all sorts of disasters befalling them, much more than I ever worried about my own DC! But I think that's because we're not the ones in control now, so we feel a kind of helplessness about them. It's not that we think our DC are not as capable as we were, not that at all. It's just that our GC are there, and we are here! And we're not the ones doing all the work, running around after them, feeling exhausted etc, so we have time to sit and think..... and worry.
You are not being extra paranoid, honestly. You're being like all of us . And it does get easier as they get older.