You're right that her anxiety is probably sincere, and not an effort to manipulate you. I have anxiety also, and have ever since I was a young child. I have a handle on it now. My husband helped me realized that I am in charge of my mind and the thoughts I entertain. It was hard, but things got better for me the more I acknowledged that fact and took control of my own thoughts instead of letting them go wild. When I have a scary irrational thought, I acknowledge it (just trying to ignore it doesn't work for me), then dismiss it as irrational and not based in reality. I visualize myself dismissing it, and then I replace it with reassuring thought or a pleasant memory. Then I remind myself that I am in charge of my thoughts and not the other way around.
I have always had anxiety about death, and I remember once when I was around 8 I was so worked up that someone might die in my family that I was sobbing uncontrollably. My dad tried to comfort me to no avail, and tried saying no one would die. What finally worked was when he said, "Honey, are any of us dead right now? Or are we all okay? Deal with that if/when it happens, but until then enjoy us while we're alive. Why waste time feeling the pain when it's not your reality right now?" It brought me back down to reality. Oh, yeah, everyone's okay, so why I am crying?
Express confidence in her ability to control her mind. It took awhile of my husband doing this before it finally sunk in with me (I just thought he was mean at first because he was being tough instead of coddling me). I was being a victim for too long and feeling like I couldn't choose any differently. I wasn't able to change until I accepted responsibility for my thoughts, so try helping her see that she's in charge. And have her attack the thoughts head on. Like, What's the worse that could happen? And how could I deal with that if it does happen? And how likely is it that that really would happen? Now dismiss the thoughts. And then she can try to focus on pleasant thoughts or meditation or guided relaxation.
I'm not like this on purpose, although the negative thoughts I sometimes allow myself to dwell on contribute to it (I don't watch the news anymore for my mental health!), and your daughter probably isn't intentionally like this either. Err on the side of empathy with her, but don't baby her or she'll never change. Feel and express sincere empathy that she is experiencing these emotions and thoughts that seem beyond her control, but encourage her to take responsibility for what she allows herself to dwell on. It is SO HARD ton take control of your thoughts, but possible. Be patient and comfort her, but also be firm. It's a delicate balance to find.
I don't like to be this intense, which is why I have put forth tremendous effort and improved, and still try to. This might be something I struggle with for the rest of my life, but it gets easier and easier. You get good at what you practice at. I was getting really good at having anxiety there for a while :) But now I can dismiss panicky thoughts on my own pretty quickly before I get paralyzed with fear. It feels so good to overcome a weakness like that and to learn to be strong and develop good coping skills! I feel so confident in myself and my ability to overcome other personality flaws, so I'm glad I faced this problem head on and created a positive history to draw on for myself. I used to always have to wake my husband up (the most common time I experience anxiety is if I half-wake up in the night and my mind is not completely aware). So you may need to help talk her through it and bring her back down to reality before she develops the skills all on her own.
Her anxiety may have reaped rewards, whether it be attention or not having to face what brings her fear, and so perhaps it fuels it on a subconscious level. It's very overwhelming to deal with these intense feelings, especially at a young age. The feelings feel very real, however irrational they may be. That's what I've learned to do--recognize and sift out the irrational thoughts. In high school I remember getting so worked up that I vomited. I felt completely out of control. Sometimes my body would shake and I didn't feel like I could do anything about it. Well, my husband let me know right away that I could do something about it and he would get exasperated with me. It freaked him out to see me out of control, something he was completely unfamiliar with. He wasn't as tender as I would have liked, but it was him insisting that I get a hold of it that made the difference for me. It's a delicate balance to find between being firm and insisting that she learn coping skills, yet remaining sensitive and empathetic at the same time.
I am completely functional and have never been on medication. Therapy is probably helpful (From what I know of them, I prefer Life Coaches over traditional counseling) but my husband has helped me enough that I haven't sought out therapy.
I would also suggest looking into the "Attacking Anxiety and Depression" program as seen here: http://www.stresscenter.com/program/
My mom gave me CD #2 to listen to, and it really helped me with positive thinking and controlling my thoughts. It's geared towards adults, but maybe they have a program for children. Browse their website for more info. Anyways, even though it's for adults, it might give you tools to help your daughter out with. Also, my parents always listened to self-help tapes in the car as I was growing up and the tapes really influenced me and I still absorbed a lot of what I heard even though I was a kid, so don't underestimate your daughter's ability to comprehend the message the CD contains. Maybe the local library has a copy of the program. Again, CD 2 was what was beneficial to me. There are probably books written specifically with children in mind.
Good luck, and way to be your child's advocate.