My teacher who is a professional doubler always told my when thinking about doubling make sure your primary instruments embouchure and techniques are completely solid. So if your embouchure is weak or changes just when you play saxophone you should not double because it could be detrimental to you sax playing, esspecially when you are thinking about going from a brass to woodwind which is a fairly major embouchure change. You must be very aware to play with the correct embouchure on both instrument. This is the advice I have been told, and it may not be all that helpful. I have never doubled on trumpet so I can not give any specifics beyond this. I am however a doubler on clarinet and flute (as far as wind instruments go), and I have noticed if I did not pay careful attention to my saxophone embouchure after I have done a night of gigging on the clarinet or flute my embouchure with sax suffers and I get very bad tone.