Isn't it funny how something that was so easy in childhood comes so hard to adults? I think part of it is the lower center of gravity and the fact that one is just fearless as a kid (bones heal faster and mom and dad pay all the health bills heh!).

200 miles is pretty new. If you want to break yourself in gently try this: sit up and spin in an easy gear. Hold your bars gently on the flats near the stem. Keep pedaling and take one hand off the bars. When you are comfortable with that, take the other hand off. Keep pedaling!