@SamCal22
For moving data between Azure File Shares across storage accounts I recommend AzCopy. The copy operation is synchronous so when the command returns, that indicates that all files have been copied. AzCopy uses server-to-server APIs, so data is copied directly between storage servers. We have a specific example of how to copy all file shares, directories, and files to another storage account. You can also utilize Storage Explorer to perform the copy, this allows you to use a GUI while utilizing AzCopy on the backend.
Hope this answers your question regarding migrating the storage portion. If you have further questions please let me know!