Here are my suggestions:
First, set up filters to send those emails to a dedicated folder and process them once a day, every day, after the rest of your email. If you don't have time one day, forgive yourself and delete the backlog.
Don't feel you have to read whole articles. I have noticed that news stories tend to sum up the story in the first paragraph, expand with some sketchy and incomplete details in the bulk of the article, and finish with another summary. I can stay current enough with the news by skimming the headlines and reading the first two sentences of any article that catches my eye. The articles I do read are the ones I find most interesting or funny.
Finally, learn to speed read if you don't already. I went to a single day course and it has been the best day training I have been on, possibly ever, but certainly in the last couple of years.