Word of Warning
While it's clear that pitch can be controlled by both embouchure and air stream, many intonation issues are actually the result of poor air stream focus and incorrect embouchure pressure. A correct approach to air stream and embouchure will alleviate a number of common intonation problems. With that being said here are some tips in approaching real time intonation adjustment.
Raising the Pitch
Adjusting pitch up or making a note sound sharper than your default pitch center is much more difficult to do than lowering the pitch, so we'll start with this problematic technique first. Trying to correct a flat note by adding embouchure pressure can quickly constrict and distort tone and even make response feel sluggish. Adding embouchure pressure can easily be counter productive, and I don't like to approach raising intonation in this fashion generally (DISCLAIMER: I'm sure I do it in a pinch or even in a very minute way on the regular basis without even realizing it).
Adjusting the pitch with air stream is also possible, and much safer than using embouchure pressure to adjust the pitch upward. There is more flexibility in raising the pitch before you start to introduce strain into your technique degrading tone and response.
Honestly, when I find myself playing flat, I usually just push in the mouthpiece because the normal result of trying to regularly adjust intonation up is degraded tone and response.
Lowering the Pitch
Adjusting the pitch down is generally easier and less risky in general. You can lower the pitch using embouchure a fair amount without distorting or constricting tone and slowing response. That's why most embouchure driven saxophone inflection (bends, etc.) deal with the pitch center and below it rather than above it.
Lowering the pitch using the air stream is also very possible ,and it might be equally as flexible as embouchure in that regard. Again if you lower the pitch too far by either method you will find that your tone and response suffer significantly.
Combining Embouchure and Air Stream Flexibility
In any kind of timbral or pitch adjustment it is most important to focus on tone quality. If I can get a beautiful tone I know that my playing will also feel very responsive, and I'll be playing at my best. I find that both of these methods, adjustment using the embouchure and the air stream, can combine when adjusting pitch in a way that allows the most flexibility and preserves the best tone quality and responsiveness of my playing.
When I do adjust pitch whether for intonation's sake or for some inflection or other I simply do whatever needs to be done without sacrificing tone. I have practiced and do practice both methods of pitch adjustment. That practice along with my focus on tone quality allows me to find the path of least resistance to adjusting intonation, or in other words, the best technique that will allow me to play and sound my best.
In terms of what to practice, I recommend practicing adjusting intonation down both with air stream and embouchure. If and when practicing adjusting pitch up, do it primarily with air stream, and even then only in very short spurts as that kind of practice can easily introduce strain into your technique. The simplest form of practicing these techniques is a slow pitch bend away from the pitch center and then returning to it. Good luck!