Hi Maria, Oh my... That's frustrating. When I click on one of those links, I get a full NYT page in tiny print also, but on the far right are the + and - buttons to enlarge the print to readable size. I also get a white overlay with the bright suggestion to "Share" the page via Email, Facebook, or Twitter, so it's hard to believe they don't want a non-NYT subscriber to read a single page. How else are they going to lure you in?
That said, when I used to read the NYT on Windows, I had lots of trouble with the browser; it wouldn't recognize my subscription and kept giving me the "so-many-free pages" line, and then it would cut me off after viewing a few pages. I say this because you shouldn't be afraid of using the "free" page option; it won't bite, that is if you can enlarge the print via the buttons or, maybe "alt" and the mouse, or however you enlarge print on your monitor. Since I changed to Google for browsing, I've had no trouble. Firefox is getting good press, but I can't speak to that browser.
I would try accessing with a different browser than whatever you're using now. Hopefully, somebody else reading this who isn't a subscriber can tell us if they are having the same trouble viewing. If I get over to our local library today, I'll try on one of their computers. However you access, I think you would have to enlarge the print.