I am more worried about your headaches than the dog, quite frankly. You ever speak to a doctor about them in the past or is this something new?
Used to be people could have a difference of opinion and still have respect for one another. We have lost that. Very sad.
I was thinking maybe you could plan on being there when the lady picks Lily up after the surgery. I would have a few questions for her as well, like where is the dog going to sleep and spend most of her day? It's too cold for Lily to spend the night outside. Also, why does the lady want the dog? I wouldn't try to put her off, because I think things will be better for you if Lily is placed in a good home, but your peace of mind is important too. Maybe the lady is a little paranoid about having people know where she lives (gee, don't know anyone like that!

). Offer her your phone number and tell her you'd like an update in a few days. You did a lot of work with Lily, socializing her and teaching her a few manners, but I think you've done all you can for her, and it's good if she can move on to a place where someone will be able to work with her. You just need a little reassurance from this lady to put you at ease. Tell her that. If she's a good person, she will understand and give you some grace.