macOS Sierra 10.12.6 not saving Screenshots
I figured it out. I have the latest mac, that comes with the Touch Bar. So I noticed that when a press command+shift+4 (without the control, so it should save directly to the desktop) my Touch Bar shows this:



So, as you can see, it was saving to the Clipboard, instead of the Desktop (I probably did this by touching the Touch Bar unintentionally). Then, if you did everything that I wrote at my question and nothing happened, probably this is your problem too. 

How to solve this:


Press command+shift+4, so your Touch Bar will look like the image above.
So, all you got to do is change the option that is selected at the Save to: section to Desktop.


Doing this, both command+shift+3 and command+shift+4 will now save your Screen Shots to the Desktop.