> Would crampons be an effective alternative to an ice axe?
Well, I'm only an early summer snow person and no mountaineer, but my answer to this would be a big NO.
Crampons can be helpful to keep you from slipping, but do nothing from there (and can be very dangerous when sliding). The ice axe is there to prevent slips (support/self belay) and stop falls once they start (self arrest).
Crampons are good for hard, icy conditions, which are rare in the High Sierra in early summer - most snow is well-consolidated, and softens up very quickly in the morning. I've carried and used them before, but typically just for descending hard, icy snow off a pass first thing in the morning; if I didn't have them, I just would've timed it differently or waited a bit for the snow to soften up in the sun.
As always, an ice axe is little good if you don't know to use it, and have practiced the skills.