Hi there :)
Can you try this for me please and let me know the results :
Can you go into settings /manage applications / chess.com
clear data and force stop, then un--install and go back onto the the MarketPlace on the Android and re-install the app.
Then try logging in again. You should only need to enter username and password once.
I just upgraded to an android from a blackberry phone (not sure its and upgrade!) anyway the site will not let me log in with the phone.. it just goes back to the log in screen over and over? any help appreciated!
Mike