Go to Appearance > Widgets and at the top right, click on Screen Options.
See if the link says Disable accessibility mode. If it does, click on it, and it the Add links should disappear and you should be able to do Drag and Drop again.
If this doesn’t work, it may have something to do with a plugin you have installed. You may have to disable all of them and enable them one by one to see which one may be causing this issue.