Note: there is still an intermittent problem with this in that sometimes a fragment of the ad page appears as the article, and the article itself is loaded as an inline link from the ad page.
I have concluded that this occurs when the NYTimes website stubbornly serves the ad page again even after a 5 second delay and a page reload.
Interestingly, this only seems to happen when there are multiple downloads taking place simultaneously.
The recipe I submitted has simultaneous_downloads=1 and this seems to prevent the problem from arising.
If anyone encounters the ads slipping in with simultaneous_downloads=1 please let me know. Otherwise I'll assume the slight increase in runtime from using simultaneous_downloads=1 is a reasonable solution to the problem.