Google Takeout / Transfer
Google Takeout is way to transfer all google apps data (ie docs, sheets, slides, emails, youtube videos etc.) Your district's google admin does have the ability to enable/disable takeout. If yours allows it, the resources below will help:
Within Google Takeout is an option to transfer all files. This is the easiest option for a person who is only concerned with transferring Google Drive files and Gmail items. Directions are here.
That may be all you need, but here are some other resources too:
Go to google.com/takeout
Select which services you want a copy of. Click Create zip file. An email gets sent to you unless you're downloading a small file size amount. (sometimes takes hours and I would check all email folders like promotions/updates folders. The email directs you to a download button so that you'll have a zipped file of all your stuff. If you have a large amount of files, you may get different parts. (My drive was so large it came back in five separate parts)
When you go to your new/other google account check your settings to be sure that they are set to convert files into the google files editor format. Then you can click new--upload file--then direct to the folder and they come in. Be sure to check though, because it will upload all the files you were the owner of, but could miss some that you were just a person that was shared a folder from someone else. For those you need to go to them, share them with another account, then go to file--make a copy so that you are an author of them and then have full control to share them as you wish.
If items are stored in a Google Team Drive, items within do not have an owner. You would need to make a copy of any items you need in the future.