Ok, I hope I can get the tears out of my eyes long enough to post

I think you should knock on their door and gush about how much she reminds you of a previous dog you had and PLEASE could you have her. Just be really nice about it and don't tell them the real reason you want her. Just say you feel so attached and bonded to her and would love to be able to have her. Tell them they can still visit her. I think if you take a very nice approach, they will be more willing. They might not even want to bother with her and will be glad you offered to take her. I hope so anyway