Welcome to the cafe! Great to have you here.
This response is not about me. I just hope that it may encourage and inspire you to use your classical sax training as a great 'grounding' to develop your own (freeer) 'musicality' away from written sheet music.
I've only ever played tenor sax so members who play multiple saxes are much better qualified to advise you on embouchure on tenor vs. alt. I may well be wrong but my gut feeling is generally that 'the lower you go in the sax family, the more relaxed your embouchure can become'. Mpc tip openings and reed strengths remaining (relatively) equal.
FWIW, I believe that 'learning classical sax' first and then learning to 'improvise away from the comfort zone of sight-reading' is a whole lot easier than the other way round! I've only ever had 2 sax tutors, both classically trained. I initially (aged 40) played for 4-5 years in a 'Wind Orchestra' (light classical music).
Since then, I've switched to playing in Big Bands, including - at times - 'free solos'. Although Big Band (swing/jazz) music typically has a different rhythm and dynamics to classical music, it's a no-brainer that Big Band musicians have to read and play music just as precisely as classical musicians. So I've always been grateful for my 'grounding' in (light) classical music. This helped me enormously in transitioning to Big Bands.
Many 'suggested' Big Band sax solos are written out. But gradually I decided to develop my own solos. Especially for the written-out solos that I found too difficult to play 😉. And also for written-out solos for which I felt: 'this is really not my (emotional) 'playing style'. So I came to rely more and more on my 'musicality' to (gradually) figure out what worked and what didn't. In other words, I gradually learned to play solos more from my heart (what I wanted to express) than from my head (playing what was 'suggested').
I've participated (on sax) in 2 10-week 'jazz improvisation workshops' which helped me better understand 'the form' (structure) of music pieces and improvisations (build up, peak, decline). Utilising (in jazz) 'blue notes' to create and resolve 'tension' in improvisations. I've also had a couple of great 'guest MD's' who've suggested that (amateur) soloists stop 'thrashing around' to (only) create variations on the melody and also just improvise on the rhythm.
For the past 6 months, I've been playing 'fills' and 'solos' in a rock/pop band with 'chords' as the only guideline. Which (playing in - for me - completely 'new' key) has been a challenge. But I've thoroughly enjoyed 'the freedom' and I've also learned a lot.
I still remember someone's comment at the cafe on my first attempt (15 years ago!) at 'improvising' to a 'Blues of the Month' or 'Song of the Month' song on the cafe: "keep at it, you just need to make more mistakes and learn from them". Good advice
Best wishes,
Mike