Instead of going to the New York Times website, get it through your library. You will avoid advertisements, be able to search for articles with ease, and get the full text of the newspaper all the way back to 1857!
Along with the New York Times the library subscribes to hundreds of newspapers from around the world, including the Wall Street Journal, Washington Post, The Toronto Star, Los Angeles Times, and many more.
To access the New York Times, click on Journals by Title from the Library's homepage. Then just type in “New York Times”, and select the database that contains the dates you're looking for.