There is nothing wrong with the clocks. They are adjusted to reflect lag in your internet connection. https://support.chess.com/customer/en/portal/articles/1444849-why-did-the-clock-times-suddenly-change-the-clocks-seem-broken-
My clock runs during oponent's move!

You don't say how you are playing on the site but there have been issues where the v2 web client doesn't get a clean connection to the game and your move isn't ever received by the server, or if you are black, you never receive white's first move.
If you find yourself waiting very long, the time depends on the time control, you can force a refresh and usually that will get the connection going again. In such an instance, the client doesn't show that you aren't connected.
The suggested solution to the problem is to switch to v3. If you are on v3, you should open a ticket.
https://support.chess.com/customer/portal/emails/new

The same happens with me. Although the connection is shown as full and no lag, my time keeps running out even after I make the move. Tried changing internet connection as well but to no avail. I have lost so many games because of this. Not able to even offer a draw in such cases. Only after the time runs out is the website showing me the move made by the opponent. Not fair at all for me as I lose so many points in this case. Chess.com please do something.

I lose on time but have over 2 minutes left!!
Your opponent has 2.24 minutes left. Your clock shows 0.00

I have the same issue. It's very frustrating, it happened several times now. It happened in the android app.
Please chess.com save my interest in this site. I had suffered several injustice in my plays. My time keeps running when an opponent's to move. Before I know it time trouble denied me sure wins. It happened when I switched to the New Chess.com and it continues after I returned back to the old. Please help me my strength is going down.