I'm giving away my secrets here, hopefully this is buried enough that NY Times won't see it and won't change. :)
1) use Firefox
2) use NoScript, don't allow Javascript on NY Times or any other domains it pulls in
3) set Firefox -> Preferences -> Privacy -> Clear history when Firefox closes -> clear all items except browsing history
To access NY Times (or really any website) first QUIT Firefox. Then restart. Now there are no cookies, caches, etc in existence. Then when you go to nytimes.com, all those other domains don't see any existing cookies. It's all new. Let them track you all they want.
When done reading, quit Firefox. Now none of the domains mentioned above can track you via cookies. Their tracking has accomplished nothing. They've all seen your single session on nytimes.com and nothing else!
Side effect from using NoScript is no paywall limits on number of articles.
One possible complaint is that I clear caches every time I exit Firefox. But I never notice any delays when starting, using a decent speed connection means the caches reload quickly. Also, I happen to have a knack for remembering passwords, so I don't mind re-entering on those sites that need them. But nytimes.com does not need a password in the scenario I just presented. (BTW I rarely re-use passwords across different sites).
I can still be tracked by IP address. Not much I can do about that other than force Comcast to give me a new one every once in a while. But that's merely a palliative, since a determined tracker can probably reestablish who I am from other information (c.f. panopticlick.eff.org).
My next step will be to get Squid running on my firewall, to help sanitize my HTTP headers, etc. But I've been meaning to do that for at least 10 years and haven't gotten around to it yet. So my paranoia does have its limits. :)