Absolutely not! As the referring organization, you are not required to take in a pet if a Rehome adoption doesn't work out.
When a private owner and adopter sign the Transfer of Ownership agreement, the adopter agrees to notify the private owner first and foremost, if the adoption does not work out, here is that excerpt:
"If at a later date I am unable or unwilling to keep this Pet, I agree to first immediately contact Owner and give them the option to reclaim the Pet at not charge. If I ever need to rehome this Pet, and Owner cannot reclaim this Pet, I agree to try in good faith and using my best efforts to find a new home for this pet while keeping him/her in my home, and to always act in the best interests of the animal."