Can't open Screenshots, can't deleted them also. Serious problem.

That is strange.

If you select one on the Desktop and tap the Space bar, do they open up in a "Quick Look" view?

How about the in the Preview Pane in a Finder window?

If you Get Info on one of them and scroll down to "Sharing and Permissions", do you see that you still have Read & Write permissions?

I fixed my own issue with ScreenShot by going into Activity Monitor:

Quit an app or process in Activity Monitor on Mac

1. In the Activity Monitor app on your Mac, under the Process Name list, select the app or process you want to quit. An unresponsive process is marked with (Not Responding). ... 
2. Click the Force Quit button ("x" in a circle) in the upper-left corner of the Activity Monitorwindow (or use the Touch Bar). This gives 2 options Quit or Force Quit.
3. I was able to quit the application - as opposed to Force Quit. Then I could open ScreenShot App again.

I have the same problem, I opened Screenshot, clicked on video and now I can no longer open Screenshot - getting the error message mentioned above, and the application shows up in Activity Monitor using up 15 - 20% capacity. I cannot shut it down. I checked the suggestions above and was able to see that I do have Read/Write Permissions. Please help. Thank you.