The feeling comes and goes for me. Sometimes (often, tbh) it feels completely hopeless. But focusing only on the bad and not being able to find the good, or to let go of the despair, leads to burnout - as you already know - and burnout is where you can, unfortunately, get the least done.
I'm not really good at knowing how to get out of burnout. My whole life is really immersed in "the bad news", so it's hard to not be around it. I do think there's a lot of value in finding good news outlets (like Reasons to Be Cheerful or Grist's Beacon newsletter) and, contrary to how it may feel or what people might say, it is healthy to disconnect from the bad news sometimes. You have to put on your own lifejacket before helping anyone else. Do the things that make you happy and don't read the news or look at your phone, if you can.
As for doing things - it's helpful to pick an area of focus. What are you most passionate about? What's the most pressing issus to you? That's where you should put your energy, as much as you're able. That could mean volunteering, doing research and contacting your representatives, spreading the word about the issues, or finding little ways you can make a difference. My issue is environmentalism, which means that I try to focus on my individual actions as a form of self-care - buying less, buying secondhand, reducing food waste, composting food scraps, contacting my reps, spreading the word. Find your little things that you can affect; those are the ones you can look at and be proud of, and let that heal you.