Lots of woodwind doubler horror stories have to do with quick switches to flute or piccolo. (“Twenty minutes of hard-driving R&B tenor saxophone, then two bars to switch to flute and enter pianissimo in the third octave…”) Doublers in this situation often beat themselves up about perceived deficiencies in their flute embouchures, and commit to even more hours of Trevor Wye, but never quite seem to solve the problem.
While daily work on the flute embouchure is crucial, as is a good warmup, I think often the real problem is the reed embouchures. If playing clarinet, saxophone, or double reeds is leaving your embouchure too tired, tense, or numb to play the flute at your best, then consider improving your reed playing. Adjust your tone production to be less tense, adjust your setup to be freer-blowing, and adjust your mindset to be focused on efficiency rather than muscular effort. Keep up the flute lessons, but touch base with good reed teachers, too.