If I'm auditioning for jazz for high school band and I play the flute, and my audition piece is for saxophone, do I play my key signature or the key signature on the sheet music?
It's not a dealbreaker for a high school audition. Transpose it so it comes out 'right' or play it as written. Maybe state which you're doing. But they want to hear you PLAY, they aren't fussed what key you do it in.
But there's another issue. Yes, flute is used in jazz bands. But almost always as a double with sax and clarinet. If you can ONLY supply the flute bits, you're going to be doing a lot of sitting around!
As a clarinet and saxophone player, I learned to transpose on sight as I would sometimes play music written at concert pitch e.g. the vocal line above a piano score. If playing alone, I could just play the written notes but if the piano was also playing then I would have to transpose. In the early days, I would envy players whose instruments played at concert pitch such as yours.
You are in the slightly unusual reverse position, playing a non-transposing instrument but aiming at a genre in which transposing instruments are common. It might help if you could demonstrate the ability to transpose on sight so the arranger does not think: "nice, but I don't want the fuss of writing a transposed flute part". Also, playing a saxophone score could give you problems with the range, It will often go too low. Will you play consistently an octave higher or just some phrases? Showing the ability to cope with that may also help.
Saxes always play off the dots that are transposed. Either Eb or Bb. If your piece is a solo, in matters not. It will sound fine to all except those blessed with absolute pitch - it's a good idea to ask if there are any, and apologise!
Flutes play at concert pitch, so their charts are written at concert pitch. If the audition involves others playing at the same time, better to find a piece written in concert pitch, as then you'll be playing in the same key, soundwise, as everyone else. A good thing.
As Laurence says, a lot of band players double (treble?) on clarinet and flute, but the majority of the time is either sax or clarinet, so it's maybe time to consider adding to your armoury?