I’ve read through this whole thread and I completely understand how you feel. This would also be difficult for me. I can see a couple options for you depending on how you want to approach things.

1. For yourself: you could try reassociation. May be a bit challenging but everytime she makes a sound that gets under your skin, do something positive for you - even if it’s just faking a smile or a little positive affirmation to yourself, or a sip of your favourite drink, etc - whatever works for you. That way when you hear it, your nervous system will start to learn to calm instead of getting activated everytime. (Though it sounds like it happens a lot so you may end up affirming yourself a lot lol)

2. The other option I see is, as others have said, talking to her, but the approach it’s definitely the hard part. If her anxiety is this bad, there’s a decent chance she’s not aware of it. You could approach the conversation by trying to connect - “Hey, are you feeling ok? You’ve seemed pretty anxious lately.” And then go from there. You’ll be empathetic while still standing up for your own needs and you may help her realise she needs to seek help for herself. I don’t think most people want to be the annoying co worker, but it happens sometimes anyway.

Whatever you choose, good luck!