I was having problems with this too up until recently. Not so much on playgrounds, but in the rec league I play in I was consistently missing lay-ups in traffic in the half-court and on the break (2-on-1 situations - made me begin to pass up to a teammate no matter what in these situations).
But I think Brandon Roy (
) knocked the nail on the head in his first point. I think I actually read a post on here and realized I had to have the right mentality when going to the hoop. I realized that I had, not neccessarily been afraid of contact, but avoiding it (part of being a more finesse than power player) and not going up agressive enough.
I changed my mindset to 'I'm going to score, or at the very least draw contact and get fouled'. Finally starting making them in traffic, and like that video in the link, using my body to protect the ball (which has lead to a ton of free throws), and I also found out that only the most determined shot blockers will actually truly contest you - guys that aren't girfted tend to just look to stop you from getting the 2 and fouling or just give it to you - particularly on the break.