Do you have direct access to your database via PHPmyAdmin, or something else like that? If so, could you go into the table exp_sites and look for the field site_system_preferences. If you look at that data, it might look intimidating, but all it really is, is the site preferences. In amongst all that you’ll see something that says
"admin_session_type";s:1:
That s:1 might also be s:2. If it is anything other then s:1, could you make it s:1 for me? What we are doing is setting the control panel to allow login with cookies only, and not sessions.
Give that a shot and let me know what happens.