Do I need to do anything for the files in my H: drive or other shared network drives?

No, the network(shared) drives like the H: drive will not be affected by the O365 migration. Only files on OneDrive (the cloud) will be migrated to the Shared Tenant.

OneDrive files will start being migrated on Friday, June 7^th. If you need to edit any OneDrive files after June 7^th, you will need to make a local copy of those files until after the migration.