Can't drag to move object. (Only arrow keys work to move a layer) Whats wrong?

Which version?

Which system?

Have you already tried a different mouse?

When was the last computer restart?

Can you do any functions with your mouse? If the problem is limited to what you've described then you might try deleting your OOBE file.

To do so:

For Macintosh Users: The User Library folder in which Illustrator’s OOBE files are stored is hidden by default on most Macintoshes. To access it make sure that Illustrator is closed and click on the desktop to launch a Finder Window (Command-N).With this window in column view follow the path User>Home folder (it’s the folder with an icon that looks like a house—it may have the user’s name rather than “Home”) and click on the Home folder. With the Option Key pressed choose Library from the Finder Go Menu. “Library” will now appear within the Home folder. Within the Library folder delete the file at the end of this path: Application Support\Adobe\OOBE\opm.db. 

For Windows Users: On Windows 7 and above these files are hidden. To find them go to the Control Panel and open Folder Options and then click the View tab. Then select “Show hidden files and folders” or “Show hidden files, folders or drive options” in Advanced Settings. Then delete (or rename) the folder at the end of this path: C:\Users\[User Name]\AppData\Local\Adobe\OOBE.\opm.db.

After deleting this file you will have to sign in again to your Creative Cloud account.