The PDFs you are uploading / downloading are being transferred using data plane APIs. The migration from Classic --> ARM is a control plane operation.
If you have any tooling interacting with the classic control plane, the Validation phase is there to give you time to retool and point those updated tools to ARM. If you don't have any control plane tooling (e.g., you created this account a long time ago and now only use the data plane), you can move immediately to ARM.
Migration is a metadata operation. Not a data movement operation. So, while it appears you have two storage accounts during the Validation phase - you in fact have one account / one data store. But it now has metadata in both Classic (ASM) and ARM. The Commit phase commits your metadata migration. Throughout the migration process, there are no impacts to the data plane - it remains available for reads and writes the entire time.
Please note that any SDK <v12 is formally retired. This SDK and the other one you mentioned are both data plane SDKs and do not interact with the control plane. The migration of any accounts you have from Classic --> ARM would not be impacted by operations coming from these SDKs.
Hope that clarifies things. If not, please let me know what else you need.