Ensure that the ADF has the correct permissions to access the storage accounts. Check the Access Control (IAM) settings for both Blob Storage and Data Lake Gen2.
Verify that the storage account's firewall settings allow access from ADF's integration runtime (IR). You might need to adjust the networking settings.
Check the ADF activity logs for any additional error messages or details related to the timeouts. This can sometimes provide more context about what's going wrong.
Use the "Monitor" feature in ADF to see if there are any insights into the pipeline runs or activity executions.
If you're using a self-hosted integration runtime, try switching to the Azure integration runtime or vice versa. This can sometimes alleviate connectivity issues.
If the issue persists after trying the above steps, it may be beneficial to raise a support ticket with Azure Support.
Provide them with the error messages, logs, and a detailed description of the issue.