Unintentional sin

It amazes me how much space is given in Leviticus to unintentional sin. Over and over it says "If a man sins unintentionally and later finds out about it . . . "Always, a sacrifice was required to cover that unintentional sin.

As a Wesleyan-Arminian, I tend to define sin as "a willful transgression of a known law of God." However, it's obvious that the biblical definition is much, much broader. Wesley, of course, did say that even our unintentional sins are in need of Christ's atonement, but it seems that we always cut ourselves some slack in that area. It seems, though, that, if we lived more carefully, more consciously in the presence of God, even unintentional sin might not be so frequent.

I often feel that I am not sufficiently contrite for known sin. Would it be possible to live so close to God that even a new awareness of a past unintentional sin would send me running to his throne to plead his mercy?