I suspect that you mean the /home/.var directory, as opposed to the /var directory at the root level. I will proceed under this assumption, please correct me if I'm mistaken.
As an aside to the topic at hand, these similarity between these named directories exists as an attempt to somewhat mimic the filesystem hierarchy of the root partition under the user's home directory. There are similar examples as well, such as /usr/share and /home/user/.local/share.
The reason why the /home/.var directory might be full is likely due to the use of a common package format known as Flatpak. This is the default package format used by the Software Store, and you're likely installed most software like this even without realizing. As Flatpak packages are designed to work on a per user basis, these must be installed under the user's home directory. Otherwise, the typical location is at locations such as /usr/lib, closer to the root of the filesystem.
Unfortunately one downside of Flatpaks is that they take quite a bit of storage space. Considering that drives are very cheap these days, it's increasingly rare to run out of space during updates but it does happen. One way of working around that is using a separate drive for the home folder, that you mount at boot time.
Despite an upgrade feature being offered, I personally like to do a fresh install. This forces me to consider things like backups and gives me a clean start. If you are willing to go this route I would very much encourage it.
Although, this means you have to re-install all your programs again. What I do to avoid that is automate the installation of programs that I know I will use, either using simple shell scripts or more specialized tools like Ansible (but that's a topic for another day).
Regarding backup of application data, you have a couple of options:
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Most software provides a built-in mechanism to export relevant data like settings. I would prefer using this to reduce the risk of incompatibilities with newer versions of the software. This is true for both Flatpak and non-Flatpak applications.
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Copy the contents of
/home/.varinto an external drive, and then copy them over once the installation is finished. I recommend you give this post a read, as the/home/.vardirectory contains non-essential files that would only make it more difficult to make the backup: How to back up Flatpak user-data and apps | Ctrl blogPlease note that Zorin OS 17 used the Flatpak version of Firefox by default, but has since moved to using the native debian package instead. This means that you are likely to run into compatibility issues if you do a fresh install now, and try to copy the entire configuration folder as is. So I would again recommend using method 1 for at least Firefox.
Try running the following command to get a size of the following directories.
sudo du -sh /usr/bin /var /etc /opt
To narrow it down further just run it again like this:
sudo du -Ssh /var/log/*
To see which directories at that location are using up more space; /var/log is a common one that can get out of hand quickly.
I find the proposed solution in this thread to be quite complete. Most programs you install will be listed under it:
Note, however, that this list does not contain packages installed as Flatpak. For that, you need to run a different command:
flatpak list
This list should be short enough that you can scan through manually to identify what you have installed yourself.