For whatever extent you want to talk about the news itself being the cause of stress (which is fair to do), I think we have to look also look at the setup of the platforms, human nature, and the culture around social media, as likely contributing factors.
Because really, however bad the news was, 20 years ago you'd be waiting for the nightly news to find out about it. Several decades before that, you'd be waiting for the following day's newspaper. Now, we're getting constant updates, and those updates may be causing a device in your pocket to vibrate and make noise every time something new comes out. We know that checking all of those notifications is addictive, and not checking causes stress. However, constantly feeling the need to check also causes stress. (human nature)
Also, we have grown to expect that everyone is constantly online, always checking all of their platforms. Speaking for myself, I get messages via various social networks, and if I don't respond immediately, people freak out and take personal offense. Even when I try to remove those apps from my phone or turn off notifications, I get angry messages from people because I'm ignoring them. (culture)
I think it's also worth pointing out that most of these platforms are not really designed for occasional use. I've thought it would be nice if you could set a time-based digest of a social networking site. For example, instead of looking at Twitter, give me a weekly digest of the tweets that (based on some criteria) I'm going to be most likely to want to read and respond to. Only update Twitter at 9am on Sunday mornings with the 25 most important tweets of the week. But Twitter doesn't work that way. It's basically built on the idea that you're always looking, always paying attention, because if you stop paying attention for a day or two, you're just going to miss things and they'll get buried under a flood of other tweets. (the platforms)
Basically, I don't think we can do much about the human-nature aspect. Realistically, I don't foresee the platforms changing because they're providing the instant-feedback that people want. In my thinking, they key would be to change the culture and expectations around social media, which would change what we want from the platform, which would change the platform.
But then, intentionally changing culture is not so easy either.