If the dog is a purebred, there has to be a rescue group out there for the dog. If the cousin doesn't get to the "adoption place" in time, please have him check with any nearby rescues for that breed.
What type of "adoption place" did the dog go to? If it is a shelter, your cousin will probably have to pay as an adopter. If he's got a connection in rescue, they often get free shelter pulls. Transport can also be done for free via many networks of volunteers.
Too bad this dog probably isn't in Texas. I've been involved in small-animal rescue for a decade, but do have dog and transport contacts.