My system crashed and after rebooting I'm left with a corrupt/broken desktop environment. I can see the toolbars and dock, however none of the icons are loading and cannot get ANY windows to display (even terminal).
I did ctrl + alt + f2 to get to the terminal and tried reinstalling gnome.
sudo apt install --reinstall gnome-desktop
However, this didn't change anything.
I also tried the fix in this post but wasn't able to find 'Linux Filesystem' partition.
First confirm that this is a system related issue, and not due to your user configuration. Temporarily create a new user account and log on there. If it also happens in a fresh default account, then the system is broken indeed.
If the system is broken, you can proceed to a system reinstall without - at this point - removing current configuration. As a well kept secret, you can reinstall the system without formatting the current system. Doing so will keep all currently installed applications and current user configuration intact.
Obviously, the target file systems have to be healthy. Before proceeding to the install, check the file systems. You could do so from a live session using the terminal, or using the tool "Disks" if that is available in the live session.
Then start the installer, and select "Something else". Manually assign the partitions the same way they are assigned in your current system, but make sure to uncheck the checkbox "Format". That way, the partition will not be erased: installed applications, user data and configuration data will remain, and system files will be overwritten.
If that fails, (or if you make a mistake) your only option will be a fresh reinstall. Needless to say that you should make sure your user documents and data are safeguarded on an up to date backup.