I assumed they just "do it."
I don't recall ever seeing any visual effects to indicate such, however it seems to me they've changed course on a number of occasions without dropping out of warp. Like if they suddenly detected something and needed to intercept.
As for avoiding collisions, don't they use their deflectors first? And then make corrections if the deflector beam failed to push the object aside?
This is what I get for subconsciously tuning out most of the technobabble.