Have you tried blocking cookies from NY Times completely in the Prefs -> Privacy -> "Exceptions" area. Also, there is https://addons.mozilla.org/en-US/firefox/addon/1243 - which does the same thing as adding to Exceptions, but with a different way of adding sites to the rules afaik.
Also, there's the whole "Private Browsing..." mode in Safari, but I couldn't find something similar last time I hunted for something along those lines in Firefox built in.
