If one drive is backing up your files properly, you will see the following status icons next to your files and folders: 

You will be able to access all files that have the status. Any files that have the status will not be accessible until you have internet.

If you know in advance you will be somewhere without internet and you want to work on specific documents that currently have the  status, simply open those documents in advance, and they will be synced locally to your computer.

When you re-establish an internet connection, all changes you have made to any files will automatically be synced back up to the cloud.

As long as you see the OneDrive status icons as mentioned above, your Desktop, Documents, and Pictures folders are all automatically being backed up. As long as you save your work in these 3 folders or in OneDrive itself, your files are all being backed up in case your computer is lost, damaged, stolen, or swapped for any reason. Any district computer you log into should automatically sync all of your files now. Note - this does not mean that your files are copied to each district  computer you log into, the files are simply available to you as if they were on each computer you are logged into.