You can purchase an Adobe Media Server license and run the product on a CentOS AMI. This is similar to running the server on a CentOS Server on your premises.
AWS does not charges when instances are shut down.
Could you suggest why you would not use the AMS AMI available on AWS ? It comes with two default EBS Volumes. You can configure them to persist even after the instance is shut down. By default, the primary EBS volume which has AMI related data is terminated . But the secondary EBS volume that has video archives or other video streaming data is persisted.