Learn to let go, even if it means walking or running at a slower pace for a few sessions. You will be getting a better workout at a slower pace without holding on than you would at a faster pace holding on.
This advice is for those without significant disability and balance issues. If you have a condition that requires holding onto the handrails, such as low vision, balance problems, etc. then you should continue to use the handrails. But consult with a trainer or physical therapist for advice on how to achieve good walking posture even if you need to use the handrails.