Hi
The first thread in this Forum is named as following:

Filesystem full - what to look for << Very useful!!!!

Please first read this thread and then you can say what is the reason that the /var file system is so full.

In my opinion the messages files are not the reason!!
(But to use the "log rolling" is very comfortable.)
The article shows you how to handle this situation.

This is my favorite command to find problematic files and directory's
du -akd /var | sort -nr | more

Best regards joerg