I have to delete all browsing data every time I login to admin

I have to delete all browsing data every time I login to admin

I moved magento site from a sub-directory (http://mysite.com/magento) to
the parent directory (http://mysite.com). After successful transfer I
noticed that my site url does not have "www" So I added it up in the
Database in core_config_data in base and secure url fields.
That is before base and secure url were: http://mysite.com
After changing it: http://www.mysite.com (added www to it).
Then I removed everything from the Var folder and deleted browser history
with cookies and all other information of chrome.
There is no problem in frontend of the site, although I am not able to
login at admin side. It just returns me to login page without any error
but with a url like this
http://www.mysite.com/index.php/admin/index/index/key/b4f1cd105c9623c8d313c41e5c59f5b1/
I have browsed same questions asked at stackoverflow and some other forums
but all suggests to clean up cookies. But I think already did that, did I
miss something or need to do something more?