When I try to sustain 2nd octave F# on my tunable Dixon alto whistle (key of A) it keeps dropping back to the 1st octave (especially when I go from 2nd octave A straight to F#). However, when I play slightly faster tunes, in which I don’t have to hold the note for more than a second, and the note is preceded by an E, nothing peculiar happens.
Does anyone know what the cause of this might be?
lack of breath control 
What Denny said…
One of the best exercises I do at least once per week but particularly often when learning a new whistle is to play a scale up then down. But I start each note with the correct breath pressure to make the note sound “in tune” with the rest of the whistle. Next I gradually reduce the pressure until it breaks or falters then gradually increase it until the note breaks or falters. I also practice intervals in the same manner. This way I not only get used to the pressure requirements to play each note correctly, but I get a feel for the controllable pitch range on each note which will prove very useful when playing airs or when a dramatic effect is needed.
If it is the actual F# you are playing, (not the equivalent position on a D whistle), then this note position (L1 second octave) can be troublesome on any whistle. I find that it often requires tonguing 
or a glottal stop/start to get it going on some instruments. YMMV.
Of course if you’re talking about the F# position equivalent on a d whistle, (R1 first octave) this should be one of your easier notes and you can quietly or noisily ignore all my suggestions above and seek another whistle, or a new mouth since one of them is defective. 
ecohawk