I would suggest to add more RAM. if you already have added some be aware that ZIMBRA needs setting adjustment to use correctly this added RAM. eg zmlocalconfig -e mailboxd_java_heap_size
If you can not add RAM maybe you can set in your /etc/sysctl.conf a lower value for vm.swappiness try to set it to vm.swappiness=10 and observe for a while if it still use ...
I've had this kind of troubles a few times on different zimbra version and it was always related to syslog.
Please check which syslog version is running (or if it's running at all) and eventually try to switch between syslog-ng or rsyslog depending on your OS version and release.
Depending on which tool you use for backups, some do support sparse files, though that support isn't as quick as the real data would suggest.
rsync -S # it works, but it isn't "efficient"
tar -S # the GNU version.
Unfortunately, my favorite backup too, rdiff-backup, doesn't have sparse file support currently. I have seen a patch to ...