The AWS documentation describes a Task Status of "Starting", however searching that in the CloudWatch results in a lot of noise. Is there a targeted CloudWatch search which will retrieve the start dates of a full DMS reload?
It appears that dms logging only uses the term "fresh start" at the beginning of a replication task re/start, however, more digging may be required to differentiate between the task start types of "start-replication", "resume-processing" and "reload-target".
Alternativly, you could use the aws dms sns event subscriptions and subscribe to alerts when a task state changes.