I think my Finch is sick and Dying? help?

Can you at least phone a vet and ask, since there is not one near you. Where do you live that there is no vet? You need to keep it warm and quiet so put it in a box or carrier and use a heating pad on one side only so the bird can move away from it if it gets too warm. Try putting a couple of drops of water in its beak to keep it hydrated. There is really nothing else you can do without seeing a vet. Chances are if this bird is sick then your other birds will also get sick, depending on what is wrong with this one.