Why participate if it leads to feeling anguish?
Because I'm an adult and not a child and I can manage a certain amount of anguish if it means helping others and changing the world for the better.
Obviously, there is a limit to what each person can do. Some people are very stoic and logical, and those people are the sort who can actually work in factory farms undercover, where they film perpetrators of the animal agriculture industry and get busted. I cannot claim I am that capable of detachment.
I am capable of of a certain level of stoicism though. I have to be in order to function within an Environmental Science program, as well as being a political activist in the past. I think I am a good candidate to be a DxE rescuer - I have enough emotion and empathy to want to be an activist, but enough strength to bear the work.
There are self-care advocacy groups for vegans. In Defense of Animals has a series of videos on proper vegan self care.
Some self-care tips for any kind of advocate or activist are:
1) Meditation or other spiritual practice. Even atheists can meditate or participate in "forest bathing" or other nature therapeutic experiences. People with religious beliefs can turn to that.
2) Knowing when to let go. We all have to turn off the news, the tv, the Internet, Twitter whatever our "engagement" is. I have struggled with this in my life, being an "information junkie" like a lot of Millenials, and a friend of mine who is a professional psychotherapist told me that even the healthiest people shouldn't spend more than about 45 minutes to an hour a day worrying about the world news or global politics, on average. We have to learn the balance between being informed and just burning ourselves out.
3) Knowing when to rest. If someone is a serious activist, that person should look for signs of burnout, like irritability, extreme tiredness, lack of empathy, minor physical illness, tenseness and soreness, anxiety and depression. It's okay to take a break from activism.
4) Eating, sleeping, etc. Taking care of one's own basic needs and not becoming obsessed with one's work or passions.
5) Not trying to be everything to everyone. Some people try to go to every demonstration and fight every cause, but ultimately one or two areas of conviction are probably best for the average person, to focus on animal rights and the environment (like me) or focus on things like racism and social justice (like many others I am acquainted with).
Obviously some people cannot participate at all. The very sick - mentally or physically - and the burnt out, exhausted, drained etc shouldn't force themselves to do so. If someone is struggling with basic life needs, like leaving an abusive partner or trying to find a safe place to live, again that's probably not the time.
But refraining from activism entirely because it causes "anguish" seems narcissistic and childish.