I'd focus on keeping deaths to a minimum, while not baiting your team, and you'll probably notice a damage spike by playing as smart as possible, shooting the same things your teammates are shooting, not taking pointless 1v1's, not rushing ahead of your team into 1v[more than one]'s.
As scout, you are probably the most heal efficient class, so playing with the beam more often will also help.
In the end, relax, you'll play your best when you're focused on making smart decisions instead of being hyper-aware of being worried about your aim, because there's so much more to scout than just brain dead aim.