- Add safe check for ShutdownHandler process existence
- Handle case where ShutdownHandler might not be started yet
- Prevent GenServer.call errors in health checks
- Use Process.whereis and Process.alive? to verify process state
This fixes Sentry errors about 'no process' when health checks
try to call ShutdownHandler.shutting_down?()
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Co-Authored-By: Claude <noreply@anthropic.com>