Hi SheliH,
It sounds like you are experiencing some issues saving screenshots to your Mac. We can help with this.
To start, do you have multiple spaces set up on your Mac? We ask because it's possible the screenshots are being saved to a different space. You can find more information about using multiple spaces here: Work in multiple spaces on Mac
If you do not have multiple spaces set up on your Mac, are you able to create and save other types of documents to your Mac or do they also disappear? We ask because it's possible the files are being saved to iCloud instead of your desktop. You can find more information about this here: Add your Desktop and Documents files to iCloud Drive
Let us know what you find out after reviewing both resources. We'd be happy to help!
Cheers!