I have been charged for RDS with zero instance running

Please use Cost Explorer to verify what is causing the charge first. Filter RDS under Service and Group By UsageType

Hello,

Is your AWS account part of a free-tier?

I would recommend to check AWS bill for usage charges or use Cost Explorer to filter by RDS and look for specific cost items.

If you feel that you've been charged unexpectedly, you can contact AWS support team

How to Address the Charges:

Delete Unused Snapshots and Backups: Go to the RDS console and delete any unneeded manual snapshots. Ensure automated backups are minimized if they are not needed.

Release Provisioned IOPS Storage: If applicable, reduce or release provisioned IOPS storage to lower costs.

Contact AWS Support: If you cannot identify the source of the charges or need assistance, submit a support ticket with AWS Support. Provide details of your billing query, and they can help you identify and resolve the issue.

You have active resources in your account that you are not aware of

To check the resources that are generating charges in your account, do the following:

1. Open the AWS Billing and Cost Management console
2. In the navigation pane, on the left side of the screen, choose Bills. The Details section shows all the charges that are being incurred by various AWS services on your account. If you see charges for a service, then that service is generating charges. If you use the consolidated billing feature in AWS Organizations, then the Bills page lists totals for all accounts on the Consolidated Bill Details tab. Choose the Bill Details by Account tab to see the activity for each account in the organization. Note: It's a best practice to check the Details section for the previous month to identify all services that are generating charges.
3. Under Details, expand each service to identify the Regions where the services have incurred charges.

Note: The Billing and Cost Management console takes about 24 hours to update usage and charge information for active resources

https://repost.aws/knowledge-center/resources-unexpected-charges