There are a wide variety of screenshot methods and capabilities. The default behavior is to show a large thumbnail in the bottom right of the screen momentarily and it should disappear as it creates the screenshot file on the Desktop. Within those few seconds during the thumbnail preview if you click the image it should open in Preview allowing you to touchup / markup the screenshot.
This sounds like a bug or 3rd party software interferring in some way. But you can work around the problem a bit till you can resolve the problem.
If you press Control + Shift + Command + 4 you will see the crosshair to select a portion of the screen. After selection and pressing Return the screenshot will be on the clipboard and it won't preview nor save to the Desktop. You can open Preview and click File New from Clipboard if you wish to mark up the screenshot or save a copy / export, etc.