I'm a native US speaker and I only know one pronunciation of "she."
I tried saying sentences with the word "she" as fast as I could, and in as relaxed a way as I could, and I still pronounced it "shee."
To confirm this, I checked an American dictionary, and noticed that as I expected, there are two pronunciations for "the," but only one for "she."
I believe there is only one pronunciation of the word "she."