ERROR: The partition with /var/lib/mysql is too full!

Check your disk space usage.

sudo df -h

You can mount large database to another partition through binding.

bind is a mount option to mount directories inside directories.

sounds like your disk is full, your probably going to need to clean out log files or make your disk larger, without knowing the finer details of your partition setup i cant advise but you may be able to mount an EBS volume on to your server and move your /var/lib/mysql folder into it then remount that volume under /var/lib/mysql and restart mysql.

once its working you should be able carry on as normal until that drive gets full

Run: sudo apt-get clean and restart MySQL process

$ sudo apt-get clean
$ sudo /etc/init.d/mysql start