I sympathize with this problem.
How do you know it's not a medical problem? This site lists several possibilities which can be treated. I once had a cat who stopped urinating after I changed it's food.
http://www.marvistavet.com/html/body_urinary_incontinence...
In my research I learned that diarrhea is almost always a medical condition. http://www.petplace.com/cats/acute-diarrhea-in-cats/page1...
In my experience diarrhea can be caused by diet.
Vet visits are very expensive and I'm not sure that I would spend the money for an exam. The only suggestion I found for containing the urine and feces is to set the cat up with a room of it's own. For me this would not be practical. You could try putting cat diapers on her but that's expensive too.
In Portland we have a couple of pet clinics whose prices are lower than most. You probably have the same in Spokane. You could also ask for advice from an animal shelter or a non-profit organization that focuses on cats.
Your memo suggested finding a new home for the cat. Who would want to take an incontinent cat with diarrhea? I suggest euthanasia. Cat rescue organizations might take the cat to avoid that. I had to euthanize my cat several years ago because she had cancer, was in pain and no longer able to eat. This was difficult for both my daughter and me but it was necessary. We missed her and mourned for awhile. However, it was not a long term pain.
If your cat does have a medical condition perhaps euthanasia is the humane way to go. Again it is probably expensive to have a vet do this. One of those cat organizations may take her and have a vet determine the cause and then euthanize if that's most appropriate than treatment.
Losing a pet is so sad but we have to do what is best for our child. I empathize with you.