Update: I did discover that I can create a new assessment and then import individual questions into it. However, this is not ideal, since I have to recreate titles, sections, settings, etc. If anyone knows how to import an entire assessment, as is, into another class, please tell me.
Thank you for your question about importing and exporting assessments between classes.
Based upon your most recent update, it looks like you found the first step that you need to perform the import/export process, creating the assessment.
The second step to copy the assessment to one of your other classes is to use the Copy Block feature. This feature only works for assessments that are in a content block, so if you haven't added the assessment to a content block, you will need to do that first. Then, you will need to click the Manage Block button (the gear icon) on the content block with the assessment, and select copy.
The following PowerSchool Learning guide provides a little more detail on how to perform this process including screen shots: https://help.powerschool.com/t5/Learning-Assessments/How-do-I-Share-or-Copy-an-Assessment/ta-p/983