I second Brenda's opinion. My oldest and me both were/have-been mouth breathers. Mine was the result of having a deviated septum, which was corrected in my 2nd year of college while my oldest daughter had adenoids and tonsils larger than an adult's because they were so infected. It's always best to have them removed as young as possible if they are causing problems because those problems will intensify and could become much worse real soon and Brenda is right about the difficulty in breaking the mouth-breathing habit. It's likely a part of what's causing the sore throat b/c the throat is getting dry.
I'd push for immediate surgery and if the doctor didn't follow through with your concerns, I'd go elsewhere.
One more thing, it's easier for kids to get over tonsillectomies and adenoectomies much easier than it is for adults so be thankful this is happening now rather than when she's grown. My husband still has his tonsils and adenoids but they give him trouble. My mother had to have her tonsils and adenoids removed about 15 years ago and she had a horrible time recovering. Even though I had to go back under because of a problem with blowing a clot (there were complications, likely associated with the start of the growth of the deviated septum yet indistinguishable at the time b/c medication wasn't that advanced back then), but after the second surgery, I was great within about 4 days or so. My oldest daughter recovered just as quickly. So far, my youngest hasn't exhibited any problems but then again she's never been one to get sick often; however, I was able to breast-feed her whereas I wasn't with my oldest because there was a problem with my milk. It's why I truly believe in breast feeding, at least for 6 weeks if the mom doesn't or can't go for the first year or at least the first 10 months or so. Those 6 weeks give the child antibodies that they don't get with formula because there is no manner in which to synthesize whatever is in mother's milk into something human-made. It's just not possible.