Anxiety is when the fight or flight reflex goes into over drive when it's not necessary, soooo exactly like you said, things DO happen where the kind of thinking is useful. It takes a lot of practice to convince yourself that things ARE ok right now, you aren't in danger. Cbt is the big therapy approach to handling anxiety right now. The idea is to identify the thought patterns, like catastrophizing, that get us into the worry cycle.
Www.getselfhelp.gg (I'll check the website when I get home) is a helpful sight I find.