May be a browser related problem - have you tried using a different browser ?
Live chess problem!
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I am also using chrome and i have no problems(i have not even deleted cookies)
The only other thing I've noticed is that I have a "paste" that will not paste. The situation is even worse on other browsers. Sometimes the first game in a session is ok but then it starts with the double move thing.
It sounds like something to do with the settings on your browser perhaps ? I think the best thing you can do is contact support.
Try this link: http://support.chess.com/Tickets/Submit
Every move I make goes to the right square then goes back to the original and makes the move again. Very disconcerting. Using Google Chrome and have cleared cache and cookies. This has been happening for about a week or so. Please help, anyone, please, it's doing my head in!