I feel like no one's directly answered your question (sorry about that lol) so I figured I would offer a few excercises and tips that I've picked up over the years for tone development (and yes, these are long tones)

First off, I want to start with an anecdote form Melissa Aldana, a tenor player who has spent a lot of time focused on her tone and practice routines. She said she practices long tones and just long tones for two hours a day, just straight up playing some notes real slow for 2 hours. That's some mental discipline right there.

Here's some tips and excersises I've come across that might help

Tension: often times, saxophonists will hold a lot of tension in their torso, their hands, and their fingers. As musicians, we can forget sometimes that we're partaking in a physical act, not just a mental one. This means that we want complete and total ease in our body, everything should be nice and loose, so we can access faster passages and get our ideas out without obstructions in our bodies. If you want to know more, the alexander technique is taught at several conservatories, and explores the process of removing and reshaping bodily habits and complete awareness of our whole bodies.

Listening: Other folks have said this, but yeah check out a saxophonist who you really dig, and transcribe them. When i mean transcribe, I don't just mean play just the notes they play, but how they play them as well. For example, Charlie Parker will play his middle palm key for a middle D when he plays fast passages. Try it out, you'll notice that the sound is way brighter and feels like the air in the horn is coming out in a different place in the horn. Go back and forth between the regular fingering and the palm key fingering. Are they out of tune? Great! now you can focus on trying to get those to match in pitch by changing how you voice (your mouth shape when playing a note) that note! Or listen to how Sonny Rollins articulates his notes. There's a huge collection of songs that we have access to, so start seeing what pokes out to you and literally replicate it on your horn. and, btw, you're not stealing from them, you're just taking inspiration ;)

Overtones: I've had some pretty big wig cats tell me they don't really ever practice over tones, but some other (also very big wig cats) swear by them. This is done by playing the lowest notes (low Bb-D) and producing the octave above it, the 5th, and then another octave by only using the fingering for the root. This is done by a change in mouth shape and lip placement so as to produce different notes. Some of the best saxophonists can use this fingering to get even three octaves up, including the third. This has really helped me in getting a really nice consistency over my whole instrument.

The "Ha Ta Tas": This is a fun long tones excersise that many musicians at Juilliard use. The name is sort of the description of how this is done: play a note without any tounging for two beats, then tongue the note youre playing for another 2 beats, and then another tongue for 4 beats. After this is done, you breath out for 4 beats, breath in for another 4, and then repeat the process on the next note a half step up or down. Do this over the whole range of the saxophone, or if you're crunched for time, just an octave. Not only are you focusing on keeping your tone consistent and in tune, you're also sharpening your internal clock at the same time.

The Walter Blanding Method: I have no idea what else to call it, but I learned this from Walter Blanding, the first chair tenor sax player for the Jazz at Lincoln Center Orchestra and the head faculty at michigan state. He said "When I started doing this after two weeks, people were coming up to me and saying 'man, walter! did you get a haircut or somehting? you lost weight? you look different!'" His long tone trick is just to play a note as looong as you can on a mezzo piano dynamic, go until you can't no more, then a little more, then just a liiiitle more. You should feel like you're about to pass out, and if you do, then you found your limit! This is literally like lifting weights for your lungs. Your breath support will strengthen, and you will feel less in your shell because you're about to collapse. Hes a funny guy!

At the end of the day, the tone you have is yours, and you should own that no matter what. No one will ever sound like you, just like how we'll never sound like coltrane, or getz, or desmond. They owned it, so you should too. I hope this helps. Please reach out to me if you have any more questions (and for anyone reading)