Take a dog on your terms, not the shelter’s terms. The paperwork they want you to sign contains a waiver of all your rights if that dog mauls you or your family. Don’t sign it!
A group of dog fosters wrote to me after one of their members (named John) was attacked by a dog he was fostering. They told me they wanted to help him pay his medical bills because the group’s insurance company would not do so. The insurance company had reviewed what John signed when he started fostering the dog: the fostering agreement had a section that waived all of his rights and even the rights of his family if the dog hurt any of them. (For details, see The Upside-Down Mind of a Rescue Group at https://www.dogbitelaw.com/blog/the-upside-down-mind-of-a-rescue-group.)
As the lawyer for dog bite victims, I review a case per week involving injuries to fosters and can attest that the contracts themselves are a fatal attack on victims’ rights, preventing victims from getting justice most of the time. If that shelter or nonprofit is sincere about placing a family-friendly dog inside your house, make them keep their word — don’t allow them to take it all back by contract. If they are looking for a sucker instead of a foster, tell them to keep their awful dog.