I am feeling better this afternoon and think I have found a solution to one problem. A pet sitter works for our vet. I talked to her, and she sounds like someone I can trust. I've already gotten to know her a little at the vet's office.
I, too, have explained to the "no pet" places that my puppy is very tiny and is not allowed full run of the house. Only one person has reconsidered, but he wanted a $750 pet deposit and a $300 cleaning fee on top of that!
Thanks again for all your comments, advice, and understanding. This is the first place I go to when I need help!