If the initial effort can be big, all you really need to do is to move your API/Backend to API Gateway and Lambda (with throttling applied), then for databases, just get a dedicated EC2 instance and host stuff manually, with periodic syncs to S3 for backups.
Most of AWS services exist because people are either too lazy or not knowledgeable enough on how to set up their own stuff on EC2.
Comments
If the initial effort can be big, all you really need to do is to move your API/Backend to API Gateway and Lambda (with throttling applied), then for databases, just get a dedicated EC2 instance and host stuff manually, with periodic syncs to S3 for backups.
Most of AWS services exist because people are either too lazy or not knowledgeable enough on how to set up their own stuff on EC2.