How can I complete delete the RDS Service from my account?

Hello.

It's hard to say without looking at the bill image, but if you deleted the RDS instance and still have a bill, it's possible that a snapshot remains.
Please try deleting any snapshots by following the steps in the document below.
I think it would be a good idea to check the snapshots in all regions.
https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_DeleteSnapshot.html

In rare cases, a resource called RDS Proxy may remain, so please check and delete it if it remains.
https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/rds-proxy-managing.html#rds-proxy-deleting

Hi Alberto,

Please try this solution.

To any active RDS instances by selecting them and choosing "Delete" under the "Actions" menu. Then, go to the "Automated backups" and "Snapshots" sections in the RDS dashboard to delete any backups and snapshots. clean up any unused subnet groups, parameter groups, and security groups within the RDS dashboard.

please look at AWS Document you will get more information.

https://docs.aws.amazon.com/AmazonRDS/latest/UserGuide/USER_DeleteSnapshot.html

https://repost.aws/knowledge-center/delete-terminate-rds

Please go the Cost Explorer and change the grouping criteria to "by usage type". This way you will see in better detail what kind of usage you're getting billed on.

It's probable you just have snapshots remaining from the past but the "usage type" will show you exactly what that is.