You cannot resize partitions while they are in use, i.e., mounted. The easiest way to repartition is to run an Ubuntu session from the live USB or DVD and do the partitioning from there. In a live session, none of your internal hard drives are mounted by default, and gparted is directly available from a live session.
Still, repartitioning is not trivial. If partitions need to be moved, it can become a lengthy process and things may go wrong. It would be much easier and faster to 1) ensure that your backup is up to date and 2) reinstall the operating system, ensuring that this time your foresee some more space for the root partition. Reinstalling Ubuntu takes between 15 minutes and an hour depending on how recent your system is. Then, there is the time you need to put your data back. Moving a large partition, in contrast, can take several hours. If that does not finish successfully, you are facing a reinstall anyway.