Well, you will put more weight on the bars when you're climbing or making maneuvers, but the gentle touch is important. Your core muscle strength holds you up on the bike, not your hands. At least that's what my racing friend taught me... still working on that core muscle strength...
Try teaching yourself that and see if it gets better.
When I worked downtown years ago and tried commuting, my hands, arms and neck were in agony every day. I realized that I was so stressed with the amount of traffic, that was holding on the the bars for dear life. Are you commuting in a busy area? Maybe your pulling on bars more because of that?
Form is important when riding. Your bike may be a perfect fit, but you still need to keep your shoulders down, elbows slightly bent, core muscles engaged, knees over the ankles. It will help to protect your body.