I'd have to see you to diagnose what's causing the cracking, but I can tell you my go-to solution: overtones. Playing all the notes that can be produced on every fingering, from lowest to highest, teaches us what we have to do to produce each distinct sound. When you practice this on high-note fingerings, most of those "overtones" are actually going to be lower than the 3rd- or 4th-octave pitch. The important technique to apply is to start with an unexaggerated relaxed embouchure, support the tone well, and move from overtone to overtone solely by gently raising the larynx, as though swooping up with your voice, while doing absolutely nothing with your jaw or lips and little to nothing with your tongue. This is related to the technique of "throat-tuning" that great flute writers like Robert Dick talk about. You'll quickly develop killer control!
PS. I recommend starting with the French vowel /y/ on all pitches.