Most people write like trash if they started school after the school computer revolution. It's just about practise. I'm right handed and my handwriting is also realy bad. I started with computers much earlier than others in my class. The people in my class were pretty good, including the lefties. The English teacher was really bad but we got the explanation to that one: When he was young he was forced to switch from left to right hand. This was before the whip was outlawed as a teaching instrument...
Most people write like trash, but I am also left handed and haven't noticed a correlation between handedness and handwriting. For what it's worth, I have very good penmanship.