Adam, when I shoot I don't see my bow or arrows, I only concentrate on the spot my arrow will go. I don't shoot cross eye dominant but have heard from other hunting archers that are and they tell me t doesn't matter. Does a dominant eye matter when you throw a baseball, basketball or horseshoes?
Pat... when I shoot a bow, the only thing I look at is the target too. Same with shooting skeet, throwing a baseball, or horseshoes. Yes, your dominant eye matters!! If it didn't we wouldn't have a dominant eye!!
In normal binocular vision there is an effect of parallax, and therefore the dominant eye is the one that is primarily relied on for precise positional information. This may be especially important in sports which require aim, such as archery, darts or shooting sports.
When I started shooting skeet at a high level, that's the first thing our coach checked... handedness and eye dominance.