My suggestion is to relax your arms and upper body. As it relates to the road bike, make sure your bike fit allows for some bend in your elbows. Don't lock your elbows and relax your grip on the bars. Ironically, a death grip on your bars will give you less, not more, control. A relaxed arms/upper body will allow you to absorb the impact of bumps on the road. It will also help you feel less fatigued. When you're out on the road, try following the white line on the side of the road to practice holding a steady line.