If you run out of disk space, it’s time to get yourself a newer one; Basic rule for users. In my case, I’m just the one who’s doing the technical work behind, which is less troublesome than the user.
To save me some headache, I usually start by replacing the disk in the PC first and wiring up the old one via USB to get the data moving.
No matter what you do to move the files, it works best with linux in my opinion. My tool of choice is the System Rescue CD booted via USB stick. Now we’re starting the actual move of files. I’m using dd here as it doesn’t waste cpu time with GUI stuff:
dd if=/dev/sdc of=/dev/sda bs=64k
sda is the internal disk, sdb is the USB stick while sdc is the old disk running as a rather big usb stick there. As soon as you start the move,
sit back get out and do something else as this takes a while.
The final work would be extending the partition(s). That’s easy with gparted in case you don’t want to mess with the bundled tools of your OS.