I tried the following command:
gsettings set org.gnome.gnome-screenshot auto-save-directory "file://$HOME/screenshots
And it didn't appear to do anything. gnome-screenshot seems to ignore this command and only use XDG_PICTURES_DIR to make decisions. I can set XDG_PICTURES_DIR to ~/screenshots, which I did and it works, but I would like to separate screenshots and pictures. Is there any way to do this?