In Illustrator, I can't move objects by dragging them
If you've recently installed Pushbullet or similar Chrome apps that use the clipboard, your problem may have an easy solution: turn off the app(s), or disable its copy/paste feature, and restart the computer.
This is especially likely to work if, on rebooting your computer, the problem resolves temporarily and then relapses.
(Apparently Adobe and Chrome don't like to share the clipboard, Chrome wins, and Adobe sulks. I just figured this solution out yesterday after more than a week of workflow logjam with the same problem in InDesign.)
If that doesn't work, the next thing to try is resetting your preference setting files, but I'll stick to one solution per answer.