A laptop is not charging the battery and occasionally shuts off even when the power cord is plugged in. This is a symptom of a bad:

The DC jack is the physical power connector on the laptop that transfers AC adapter power to the system. When it fails or develops a loose connection, power delivery becomes intermittent, preventing the battery from charging and causing the system to lose power unpredictably. This matches both symptoms described - no charging and random shutoffs while plugged in.

A faulty hard drive causes data errors or boot failures, not power delivery problems.

A failing cooling system causes thermal throttling or heat-related shutdowns, not a failure to charge the battery.

A bad battery would fail to hold a charge but would not prevent the system from running on AC power alone or cause charging indicators to fail.