How can I check CloudWatch agent and shell script use which apis.

PutMetricData & PutLogEvents are the API calls used to publish metric datapoints and log events respectively. This is regardless of the method used to publish data to CloudWatch. These events are not logged by default on CloudTrail as they are expected to be very high in number throughout the account.

To understand the charges with respect to CloudWatch, read this - https://aws.amazon.com/premiumsupport/knowledge-center/cloudwatch-understand-and-reduce-charges/

Usually, third-party softwares where monitoring is done by using GetMetricData action (frequently in high numbers) causing for a surge in the usage of CloudWatch APIs can result in high charges. However, the above article details out all other possible reasons & methods to reduce future charges.

Hope this helps!