Looking for Sax-specific advice

Definitely long tones to improve tone, blend and co-ordination. There are points where if you have a decent saxophone section you wont have too many special specific warm up exercises for saxophones.

However being a section leader of the saxophones for a concert band, stage band and quartet for many years now I've found that playing F sharps in both Bb and Eb (tunings) and raising it by a semitone each time helps improve intonation over the entire playing range. 5 minutes of this is invaluable.

The Lead alto role is particularly important in a jazz ensemble with multiple saxes, as each other member of the section should match their tonal quality as closely as possibly to ensure a smooth blend and a sound that is distinctive of one instrument and not of different instruments in harmony, you know what i mean.

Warm up time is typically less for saxes than for brasses, all you really need to do is get the instrument to stable playing temperature so that tuning can happen. Prep time is longer though, as reeds come into play. This is even more true for clarinets and even more true for double reeds.

ksentine, can you explain your F sharp excersize?
im not sure i understand