Had my second flute class today, at my teacher's place over in Hackney central. From today's lesson, it seems like why I'm running out of breath quickly is because I'm blowing a massive stream of air - and I can barely support it for more than a few beats. The proper way, which still eludes me is to focus the embouchure into a tighter, small stream. Gonna take awhile as I find it quite difficult to adapt, but already I could manage about 2 bars at a reasonably slow tempo without fainting.
That said, I think there is an advantage of learning from different instrument with a different teacher - everyone has a different things to teach - one of the key points I picked up is about practicing with long tones. It's not just blowing long notes, but to figure out what is the best position for one's embouchure to be in per note.
With the saxophone, I didn't think of it this way, my understanding was "long tones" == "tone gets better". Time spent with long tones needs to be used to find out how to make that particular note as sweet as possible.
With the flute, it's finding the sweet spot with the various variables - placement of lip plate, roll in/out, embouchure etc. With the sax, in my experience it's more about the shape of the tongue inside to direct the air, as well as the firmness of the embouchure support.
This also brings up the point of doubling - the flute and sax use wildly different embouchures, and I believe that either won't interfere with the other much. Fingering wise, it's similar, but the flute seems to utilize overblowing to get to the next register, whilst with the sax it's a combination of keys and embouchure to get there.
Lots to think about and practice.