have had/have been having a headache

yes, she is. "I have had a bad headache for a week now" sounds as if you've had a single headache the entire time (which would be terribly unpleasant). I've been having a bad headache" sounds more like a series, but you'd also be more likely to say "I've benn having bad headaches" for a series. The end of the sentence sounds stilted. Try "for the past week" or "for a week now" or "for one week" or "for one week now" or "for a week", any of which will work.

I have had a headache for the past week. (The same continuous headache)

I have been having severe headaches for the past week. ( a series of headaches)

Dump "one" and "now".