I agree that clothes look too grown up.

You could google online stores with "modest" or "christian" in the search, and you may find some kids clothes that are more to your liking.
Have you talked to a dietician? You could get a family appointment, not just an appointment for her, and make a family commitment to health. She wouldn't know you actually had her in mind, but she would see support in a more round about way.
Puberty could be a big part of the problem. I remember the same thing happening to me. I hit puberty 3-5 years before all my friends, so I always wore oversized clothes to hide my curves. I was
very insecure until most caught up to me in high school.
As for exercising -
Muscle burns fat, but fat can't be burned without aerobic activity. Doing things like situps or weights will build/tone muscle, but it won't do anything for fatty areas. Walking is much healthier for joints than running, and even something like biking would be easier than walking for someone with asthma. (My hubby has asthma.) What about swimming? Swimming is great for muscle toning and aerobic exercise at the same time. Even just walking around in waist-deep water is enough resistance to get a good workout (that's what I do... I hate swimming).