Hi Whit, try to boot from a live system like Knoppix or any other live-cd.
Then mount the partition, where /var resides and you can delete files you do not need (for example old packages). If your filesystem is not encrypted, then youz might want to enlarge the partition without any loss. You can do this with the famous gparted-live-cd for example. Maybe you have enough space on a partition on another computer (filesystem must be ext3 or ext4), then you can sync all data from /var to this computer and after enlarging the /var partition back to it. With live-systems like Knoppix or TRK you can do many nice things, resueing the system or just get access to filesystems when the operating system does not start any more. Feel free to ask for more information. Hope this helps, good luck! Best regards Hans