Hi @musical_mercenary, The flute is almost ENTIRELY about tone production. Your approach to this fundamental problem dictates how well you will play in tune, whether you can successfully play technically difficult passages and whether you look forward to or dread playing exposed passages.
As a professional woodwinds player (like you, sax was first), I spent a lot of years trying to advance my flute playing by watching Youtube master classes, which were given by great players, but I did not really start to crack the flute and piccolo nut until I started taking lessons with (at different times) three fantastic flute teachers who currently or had previously played in the Milwaukee Symphony Orchestra flute section. The very specific feedback you get on your playing from even an occasional lesson with a great teacher is gold and can really help focus your vision and your practice.
A couple things to keep in mind are that flute and piccolo have very different pitch tendencies due to the bore differences. Flute is much less forgiving of leaks than sax, so make sure it’s always in good repair. To project more on the flute, concentrate on playing always with great resonance rather than blowing more and more air through the flute to force a forte. Spend a small amount of time every day on tone development exercises for ever and ever. The Trevor Wye Practice books, especially volume 1 Tone, are excellent resources.
Approach all of your doubles like they are your primary instrument, learning the specific technique of each. Take a lesson here and there. Try a few different great flutists who teach enough to know how to communicate and go back to the one(s) who give you what you need to succeed. Good luck! It’s great that you are still working to improve. It’s the mark of a real pro.