Disk space exhaustion from logs

A subset of your service instances start failing writes with No space left on device, return 500s, and some crash on startup. The host dashboards show root-volume disk usage hit 100% on the affected nodes; CPU and memory are normal. A change yesterday bumped the application log level to DEBUG to chase a bug, and log rotation hasn't kept up, the local disk filled with verbose logs and a few large core dumps. How do you triage and recover, and how do you keep disk from being a silent killer?