You may sue an animal shelter if their dog bit you inside their premises. In such cases, liability may depend on who had ownership of the dog at the time of incident.
If the animal shelter had possession of a dog and allowed the dog to escape, they may be held liable for the dog bite. During the time of the incident, the animal shelter was the party responsible for taking care of the dog. If an employee was taking care of the dog when the dog bite occurred, liability may still be imputed to the shelter.
However, the animal shelter may not be held liable if another person had already adopted the dog. Adoption contract usually transfers ownership of the dog with finality. It then terminates the liability of the animal shelter for the dog. The animal shelter may reserve the right to take back the dog when the new owner is abusive or negligent. However, this right does not affect the finality of the transfer of ownership of the dog. So unless and until the animal shelter take back the dog, the new owner is liable to the behavior of the dog.