Linux

If you can boot into terminal session, e.g. by pressing Ctrl + Alt + F2, you can try:

sudo apt clean
sudo apt -f install
sudo apt clean
sudo apt dist-upgrade
sudo apt clean

If sudo apt -f install doesn't work properly, you can create an apt-cache folder on, e.g. your home partition, assuming this is the one with sufficient amounts of free storage.

sudo apt clean
sudo mkdir /home/apt-cache
sudo nano /etc/fstab

In the fstab you specify where this directory shall be mounted:

/home/apt-cache    /var/cache/apt/archives    none    bind    0    0

Now you copy the files in place and mount the partition:

sudo cp -r /var/cache/apt/archives/* /home/apt-cache
sudo mount -a

Nou you should be able to run the fix-installation and update commands without the errors:

sudo apt -f install
sudo apt dist-upgrade

This may seem like an obvious question, but are there files you can remove or perhaps move to another drive or USB stick temporarily to make enough space to get through your updates? You should be able to do those while rootfs is full.

We can certainly delete or copy files using the terminal.

Are you sure the root is full and not readonly due to other errors? Why do you believe root is full?