Second, you'll want to clearly understand what is causing the disk to fill up. However, I'm not sure what you mean by DNS resolution and why this would affect your disk. You need to understand what is happening and address the source of the problem. Failure to do so means adding disk space and may not solve your problem. If it's not a DNS problem, I would confirm whether you're doing Exchange-aware backups. Non-Exchange aware backups don't purge the transaction log files and are a common culprit in the scenario you describe.
The simplest solution -- once you've identified the source of the problem -- is to physically add storage to the drive that contains the database and log files. This is assuming that this is something you can easily do with your server hardware. You may end up having to temporarily move some mailboxes off the database in question -- but in any decent datacenter setup, adding storage shouldn't be a huge issue.
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