If you had only / partition getting filled up with docker images or containers, and you wish to move the docker directory to other location below steps can help you ..
Change your default storage base directory for docker(containers and images) from file /etc/sysconfig/docker.
# grep other_args /etc/sysconfig/docker
steps to be taken note while moving from one location to other location:
1. stop your running containers on the docker
#docker stop <container_names>
2. stop docker service
#service docker stop or systemctl stop docker.service
Double check and confirm docker service stopped.
3. make sure you backup your current /var/lib/docker before making any changes.
#tar -cvf var_lib_docker-backup-$(date +%s).tar.gz /var/lib/docker/
4. Move your directory to your desired location
#mv /var/lib/docker /your_desired_directory
5. Create a symlink to your new diretory
# ln -s /your_desired_directory /var/lib/docker
6. start your docker service
#service docker start or systemctl start docker.service
7. start your containers
# docker ps -a
# docker start <container_names>