It could be a weight distribution issue. Your weight is probably proportioned differently on a road bike than on your old bike you had at age 10. Additionally, the headtube angle, trail, offset and track on a roadbike most likely contributes to less lax handling than your old bike as well. It just takes a bit of practice to relearn some balancing skills. One thing that helps is keeping your speed up. Another thing is to practice going slower and slower while trying to maintain your balance. Then try doing it with no hands.

Here's a picture I took a while back of me just cruising around with no hands. I'm doing close to 18MPH in this shot.