I agree with simon3142. When speaking , try to use contractions as much as possible (I'm, She's, don't, can't). This way, you will sound more natural :)) An example when you want to emphasize would be "Do not panic, I am on my way!" :))

It sounds OK to me but I would probably shorten "do not panic" to "don't" unless you really want to emphasise "not" - "I do NOT panic"

It depends on the point you're trying to get across. anxiety is closely related to something called "panic attacks". If this describes what you are expressing, then a better sentence would be, "I suffer from anxiety, but not from panic attacks." or, "I suffer from anxiety, but I don't have panic attacks."