i do. but for me, flute takes awhile to warm up on, whereas for me picc. doesnt as much. i also find that when i practice my saxophones for a while, it helps loosen up my lips for the flute/picc. i also play alto and tenor saxes and occasionally bari sax, oboe, clarinet and bass cl, and bassoon<but not as much bssn anymore>.
I'm woodwind fluent...My embouchures don't become ruined by playing the other instruments, but I've been doubling or whatever you want to call it for about 12 years. You get used to it after while.