Is there a way to adjust the vertical position of slashes so I can get rid of this collision? (attached.)
This is for a piano part that should be comping behind a horn melody, in this case led by an alto saxophone.
Yes, you can adjust them via the Properties Panel.
Ah, found it! “Slash pos.”, the very first option. 
edit:
Hmmm, that changes the position in the part and in the score, where in the score the cue isn’t shown. Is there a way to show the cue in both the score and the part?
I suppose I could just not use the “cue” object and use different voices instead.
You could use a slash voice, and set the stem length to 0.
I confess I’m surprised that slash region position propagates automatically.