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I agree with TXbandit. Try not to look down or around at circumstances. Protect your mind from the toxic thoughts. Most of them, as bad as they may feel, are just electrical impulses firing at random. They don't mean anything.
Rest is worth the effort (a contradiction in terms), to give yourself the break you need. Conscientious people will look after others first and feel guilty about sitting around "resting." But to better take of others, or a job, etc. rest is necessary. Even if it means things don't get done for a few days.
Triage the demands if possible. List what is critical first, then necessary second, and last, what can wait. When I do this, normally there's nothing in the critical column and even the necessary items can be moved around to give space for rest.
You've heard the one about loosing a bow's string? To take the tension off occasionally keeps the bow strong. There is also a wisdom proverb about taking time out to sharpen the saw. Both precepts are good for the long term efforts of our lives.
Being tired really does give place to discouragement. The one feeds on the other.
It's not about feeling worthy enough to take care of yourself, either. It's about rest. Period. It's the way we're wired.
When I read your post I thought "I know how that feels, that kind of despair. Here's someone else who understands." It gives a sense of connection, right?
Hope it helps being in touch and thanks for allowing others to reach out to you. It gives me a sense of purpose, something I don't have very often.