Exhaustion is the first step towards burnout. Burnout is exhaustion that doesn’t go away despite weeks or months (sometimes even years) of rest. You do sound like you’re on your way there, especially because you feel utterly exhausted, and like the work is inevitable. It’s not. If you died, someone else would do it.
A good way to take a break when you feel like you can’t is to become physically sick. Yes, that’s not honest, but it’s not your fault the world doesn’t respect mental strain as much as a physical illness. Your brain is just an organ. It can be damaged by stress, it can, for example, get inflamed. And just like with other organs (say, your lungs), ignoring problems will create complications and make treatment much more difficult.
So I really, really think that at your level of stress, it’s okay to call in sick with a migraine or a stomach flu (even better, as it gives you 3 days or so). Conscientious people like you won’t abuse this (you’d feel too guilty). If you don’t take care of your body, it will handle things for you. Probably by taking you out of commission for a much longer period. I know someone who ignored their body’s signals to the point of burnout, then severe depression, and then hallucinations. It took them about two years to be somewhat close to their previous level of productivity and happiness. I don’t think they’ll ever be fully there again. And you know what? That’s okay! We don’t have to be as productive as we’re encouraged/pressured to be.
Nobody has a right to give you more work than you can handle. You need to sleep. You need to eat. You need rest, which is a requirement that’s different from sleep and includes fun, relaxation, and spacing out. You’re a human with a big brain, and your brain needs sleep, food, and rest to keep going. Otherwise, just like with a car that’s run out of fuel, you’ll grind to a halt. It doesn’t matter how much you will yourself to go on. You can’t talk a car into moving just a little farther when it’s running on empty, and it doesn’t work with brains, either.
Go to bed, okay? You’ll feel better soon.