hmm...
what I'm doing when my root partition is running out of space is an "apt autoremove" and an "apt clean".
before:
root@siducer:~# du -sh /var/cache/apt/archives
4,6G /var/cache/apt/archives
after:
root@siducer:~# du -sh /var/cache/apt/archives
368K /var/cache/apt/archives
This last one empties the archive of apt, which get filled by every upgrade of the system.
If thats not enough free space after, your "gparted" offers under "Partition" the easy to use option "resize", where you can give your root partition a few GB from your /home by simply moving the borders in the diagram of your partiton. I had used this option now and then with no problems ever.
First shrink your home the amount of storage you like to give your root, eventually shift your home to the position needed, then resize your root in the now freeed space. Of course this is not working on a running system, so you have to use a live system to do so.
greets
ayla