Directing your attention is a combination of willpower and habit and external influences.
Choose what you want to do (other than "feeling sad"). When you have free time, immediately start doing this (other than "feel sad first, and then maybe do my projects later, unless I am already overwhelmed by the sadness"). If there are ways to change your environment to make it easier to focus on doing your projects (maybe put a TO DO list on a wall next to your table), do that.
Pretty much the same advice that would apply to procrastination.
Some meditation exercises may help you System-1-believe that your attention actually is under your control, to certain degree, which can be improved by training.
Check your beliefs. Is there some part of your brain that says that feeling bad about X is actually helping X? If yes, that part is a liar... or rather, it is optimizing for status, because feeling bad about X is a way to signal that you care about X, without actually helping X. So, if you can somehow help X directly, do it; otherwise recognize the status instinct for what it is and tell it to shut up. (It may be easier to tell your instincts to shut up after you have reframed them from saintly to selfish.)
Honestly I think the real answer is to start working on the state of the world.
The best medicine is to attempt to do the best you can - which usually turns out to be more than you expect - and then feel some consolation in knowing that you're doing all you can.
Note that decreasing your own happiness in tune with the perceived sadness of the world without that decrease in happiness resulting in some actions increasing the world happiness is anti-utilitarian. The whole point of the "grim-o-meter" connected to the state of the world, rather than to "local occasions" is to spur one into action on the world scale. If that casual arrow is not under your control, or is maxed out, say, by giving what you can to an EA charity, feeding the world's sadness into your own makes things worse for everyone, and decreases average happiness for no good reason...
From detach the grim-o-meter:
The quote, and the rest of the essay, seem to imply that once someone notices that their grim-o-meter is connected to the state of the whole world, they can detach it on command. Personally, I'm having some trouble, and I imagine others are too, so I'm asking the community: What advice do you have for someone struggling to detach their grim-o-meter?