First, a straight saxophone should be played at about a minimum of 45 degree angle to your body.  Some players play them almost straight out, parallel to the ground. It is not a clarinet that has a different shaped mouthpiece. Since you are playing over a marching bank, keep this in mind.

Do long tones. Dont be to worried about the fast fingers. Play the notes with a tuner and relax your neck muscles. John Coletrane would play and you could see his entire upper body expand with his relaxed throat muscles moving air. Long tones are long relaxed notes in the area you are truly concerned with.

The Soprano is a totally different animal from the alto and Tenor. It is the harder horn to play well. In my College audition years, I auditioned on Soprano for my solo piece. Villa Lobos Fantasia. It was a very challenging acrobatic feat, but the goal was to play it flawlessly and effortlessly. My other pieces were on Tenor. Yes I was a major John Coltrane fan at 18 .

Work on those long tones for a good solid tone.

Do it in front of a mirror and make sure you are not biting too hard.Roll your lower jaw to pull notes into tune.

Also....

Advice from an internet post is something everybody should take with a grain of salt.

Good Luck

If you are biting down and still play flat, maybe you should try a softer reed.  My impression of blue box Vandorens is that they are a bit hard.  I would only go down about a 1/2 strength to start out, and you will need to reduce the biting to avoid pinching off the reed.  Also, check and see if your lower lip is positioned near the break point of the reed to the table.  Playing too far out near the tip can cause intonation issues, and dampen reed response.

Tongue position is a biggie, since the volume of the oral cavity and the mouthpiece work together to make the volume of the resonance chamber around the reed, which affects the intonation.   You might try raising your tongue position to decrease the volume of your oral cavity as an alternative to pinching the reed, to correct the upper-note intonation.  Once you've got that dialed in, you will find very little adjustment necessary to play up and down the scale.

Slight necro here, but it seems the responses have not mentioned this thing that can happen on sopranos to make the palm keys uniformly flat. Well, at least, it's something that has happened to me to the extent that everything is fine up to and including the C#3, and then the palm D actually sounds pretty much like the C#3, and the rest of the palm notes are skewed correspondingly, just like the OP describes. 

The thing is, the palm D tone hole (at least on my soprano) turns out to be quite tiny, and the key clearance might not be all that great either. So tiny that it can easily get blocked by gunk/fluff or even a drop of spit^H^H^H^Hcondensation. if it's gunk, a pipe cleaner or something like it may help restore free passage. In my case, absorbing stuck fluid with a piece of paper will make the palm D sing again. 

YMMV