You should be able to download everything from your old account, either directly from Google Drive or, if the school enabled it, through Google Takeout. In either case, the Google document formats will be converted to offline formats (Docs -> Word format, etc.). Then, you can unzip the downloaded archive and upload the entire directory to the new account. Before you upload the files, open the Drive settings and choose the option to convert files on upload, if you want them in Google document format again.