The sax is an imperfect instrument. The higher the pitch of the sax (like soprano) the worse that becomes in terms of intonation/tuning. With a keyboard or good string instrument you hit the note and it comes out perfectly (if you're in tune to begin with). Playing tenor is much less difficult as you can literally just blow, key a note, and the pitch is kind of close to being right. With each of the higher pitched horns you must HEAR the note you are playing in order to play it in tune. A tuner will NOT help when you are not in front of it. You must hear the pitch accurately in order to make embouchure and body (oral cavity, tongue, throat, etc.) adjustments or you will never play in tune. Some people mistake this for needing perfect pitch...it's not. It's much more like singing or whistling where you hear the pitch you want and make it happen. You are developing a feedback system in YOUR body in order to hear the correct pitch.
If you are only reading music and strictly developing an eye to hand reaction without hearing the note you are trying to play, then you will never play in tune. The tuner is the worst thing to use as you are once again only using your eyes instead of your ears to make corrections. How is that going to work when you can't see the tuner?
If you can whistle or sing with relative accuracy then you should be able to develop the same feedback system to play any sax in tune. Long tones are always a good practice, but if you want to play in tune, intervals are much more important. There are a number of "ear training" exercises that can be found on the web, or you can also just play tunes you know that have lots of intervals where you MUST hear the note in your head and are listening intently to hear whether you are accurately achieving that pitch with a consistent good tone.
If you are "tone deaf", which is certainly a common condition, then you will never be able to play a high pitched sax in tune. Switch to a tuned instrument that you don't have to fight with or just play a lower pitch sax with closer (but not usually 100% accurate) tuning.