Only one way and you know it.

PRACTICE

There are plenty of things you can do:

1.) Obviously, listen to Jazz Sax. The best thing you can do is to find a sax sound you like and mimic it. There is no real right or wrong tone in Jazz, especially on improve, but some notables with unique tones are Paul Desmond, who famously said he wished to sound like a dry martini; Lester Young; and Cannonball Adderley just to start.

2.) Practice swing and large intonations. Swing not only helps with timing, but it absolutely helps with your tone as it forces your embouchure to change with that style. As for intonations, you probably have a book covering it, however, my personal recommendation is spread triads. If you don't know what they are, here's a video explaining them: https://www.youtube.com/watch?v=tLYqB_4xZ5o


Spread triads are fantastic for intonation and Jazz improvisation. I suggest going up and down on every note until your jaw tingles from being tired out.

3.)SCALES: I know schools tend to make these boring. However, playing atleast more exotic scales will absolutely assist in getting down improvisation. This website, even though it doesn't use sheet music, will allow you to explore more esoteric options. If you haven't I'd start off with the Blues Pentatonic Scale, Be-Bop Scale, Minor Scale, Harmonic Minor Scale, and the Melodic Minor Scale.

Website: https://www.pianoscales.org/jazz.html


4.)Mimic music from other languages and dialects or utilize another medium that uses another language fluently, such as Anime, Kdramas, or Spanish Soap Operas. You may not believe it, but the language you hear and speak has a massive influence on what you output. Absorbing as many dialects and languages as you can, will decicively diversify your improvisation. 

For more diversified Jazz, you can look towards Chick Corea, John Coltrane, Miles Davis, Dave Brubeck, Dizzie Gellespi, and plenty of others, notably in the Fusion, Latin, and Hard-Bop genres.

5.)Listen to your neighbor. Improvisation is a moving composition.Whatever your neighbor did, add on to it. Wynton Marsalis, the director of Juliard's Jazz program has gone on record saying the #1 problem he has with students is them not listening to the last solo. 

6.)Write music. You don't even have to litterally write it down. Just imagine possibilities in your head through out your day. If you want to get more into it, there are plenty of YouTube channels out there just for this. Rick Beato, Adam Neeley, and especially Nahre Sol are great, fantastic places to start. Nahre Sol specifically has a series where she learns different styles of music just for inspiration.

7.)Equipment: I'm sorry to say, but sometimes it just comes down to this. Assuming you're not a billionaire who can buy 20 saxophones without going under, I would look into atleast a Jazz reed set and maybe even a new mouth piece. Play around and explore, many companies like to use demo reed packs with one of each type of reed. I personally use Jazz Rigati Golds size 4, however, most people would lean towards a 3.

Finally avoid these:
"The Jazz Real Book" - Too many people have come out saying it's a scam.
American Pop Music - There's no problem with the music itself, the current state of it though does not support Jazz very highly and in my opinion really can harm your playing. I find that music negatively affects your thought capacity as well.
"Student or Beginner Equipment" - Unless you're renting or just starting, there's no point in this. Just go for professional if you can afford it.
Chords Only - I fell into this trap when I was younger. If anyone at all tells you only to play within the chords, they likely do not know what they're talking about. Although, it is good to do this when just starting, it's better to use any similar chords or scales around it. I would be careful though, as obviously you need to know what you're doing.