Your problems are almost positively the flute. I played on a Sky (Piece of Far East Junk) for about a year, and had the same problems. I bought the flute, and was young and stupid at the time, and later learned it was doing me anything BUT good. I then got a Selmer Prelude, which was a nice student flute, but now is doing me nothing. I recently bought an Armstrong Omega professional flute and it is wonderful.
Armstrong is definitely the way to go. Many people will advise against this, but if you have played on one, and liked it, get one. Other good brands are Yamaha, Pearl, Jupiter, Eastman, Trevor James, and a bunch of other high end intermediate and professional models that are incredibly expensive. A lot of people swear by Yamaha. I can't stand them. A brand to avoid is Gemeinhardt. They aren't much better than the far east junk flutes in my opinion.
Most intermediate models, new, are $1000+. You can get really nice used ones for much less though. My Armstrong was $500, new would be over $2000. If you can, go to a music store and test play multiple flutes until you find one you like. You might find one you like better than an Armstrong, for a better price. The people at the store can help you, and will also help set up payment options if you don't have all the money to pay for it right then.
Also, if you are upgrading, definitely get an open hole. Along with a B foot joint with Gizmo key, and off-set G. Split E mechanism is also a plus, that I wish my flute had, cause high E is the only note I have an problems with. Sterling or solid silver headjoints are a plus as well. Some say solid or sterling silver bodies are better too, but I prefer a sterling silver headjoint on a silver plated body.
As for your grandma, I know what it's like. My dad bought me a far east junk Saxophone for Christmas. He was really mad when I told him I wouldn't play it, but it was for a lot of reasons. My dad is weird. If you just explain to your grandma that her intentions were really good, and you love the flute and won't get rid of it, she will understand.
Anyway, hope this was a help. Good luck finding a great flute! And also, congrats on first chair and the solo. (: