I'm not sure if this App will work any better for you. I don't have a premium subscription, I mainly use it to browse the headlines. It's a pretty nice App. It is an ad-supported version however, I think if you are subscriber these will go away. It has clean and smooth navigation with all scrolling up and down for articles, and side to side scrolling to change articles.
You can download Apps for the Kindle Fire outside of the Amazon Store. To install Apps outside of the Amazon Store you have to set it up on your Fire. Touch Quick Settings, More, Device, and enable Allow Installation of Applications from Unknown Sources.
When downloading Apps outside of the Amazon App Store, you will see a small pop-up in the Notification Area of the Status Bar at the Top of the Fire. If you touch the Notification you will see the download in progress. After the download is complete, touch it to launch the App installation package.
http://androiddev.orkitra.com/downlo...62425004254566
Being a premium subscriber you should be able to download a mobile App directly from the New York Times which may work better.
http://www.nytimes.com/subscriptions...mpaignid=37WYW
One last question, are you sure the Fire isn't downloading the paper and storing it? Use a File Explorer like ES File Explorer or File Expert and take a look in the download folder.