Contract Technicality

My family and I adopted a dog from the Local Shelter. Due to the amount of energy he has, we thought finding him a home with a big yard would be better and posted him for adoption online pending a home visit and a perfect fit. The shelter contacted us after seeing our ad and informed us that if we took him anywhere else, that would be breach of contract. After reading over the contract and realizing that was so, we decided to keep him and get a bigger yard after our lease is up. We'd rather see him with no yard for a few months than no yard AND no family in the shelter he came from with sores all over his body. However, the shelter is demanding we give the dog back or there will be legal action taken for breach of contract. If we still have the dog, he's never left our possession and the contract was never actually broken, can they really force us to give him back? Thanks for your help