Dang man, I feel you on having anxiety talking to a girl you like. Been there before too. But don't beat yourself up too much, you recognized she noticed you too which is a good sign!
First thing is breathe - next time you see her, slow down and remind yourself she probably won't bite your head off. Smile and say hi if you make eye contact again.
As for apologizing, I wouldn't do that yet since you haven't really done anything wrong. But a casual "hey, sorry about last time, had to run but glad we could say hi now" could break the ice.
Also see if you can find a reason to casually chat - ask a work question or comment on something happening nearby. Small talk is your friend here til you get more comfortable.
Don't put too much pressure on yourself. Just say hi when you see her and let it progress naturally from there. You got this man! Take it slow and see where it goes. She might be interested in getting to know you too.
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You're not in love, you're just infatuated. Get a hold of yourself and leave the girls at work alone, unless you want to be looking for a new job.
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You don't.
If she's not approached you, then she's either being creeped out, or not interested.
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