Try logging out of the site on the computer that is not working. Then, using internet explorer (firefox sometimes won't write the session cookie correctly) clear your cache entirely, go to everquest.allakhazam.com, then press Shift+ refresh. now log in if login is successful you should get the session cookie.
If that doesn't work, you might need to run disk defragmentor. the temporary internet files location can sometimes get fragmented to the point that it has a difficult time with new cookies. also make sure your temporary internet files storage size is set large enough that it isn't just dumping all new cokies for lack of space. A few hundred MB should be sufficient for most people.