A 5 or 6 year old Yorkie has plenty of life left, and a rescue can only guess as best they can. You can't fault them for that.
Is she spayed? Her dribbling issue is probably related to fear/submissiveness. Bending down over her may frighten her, so I would try crouching near her and calling her to come, then praise and/or giving her a treat every time she comes to you. Lots of positive reinforcement and patience will go a long way with your little one. Recognize that if she came from bad circumstances she may not be socialized, so to her everything is new and scary, as well as confusing. 1 week is not nearly enough time to expect her to be housetrained, so just be patient and consistent with that. And don't ever scold her for accidents. I wouldn't even scold her if you catch her in the act, just quietly move her where she's supposed to go. I gave step-by-step housetraining instructions in this thread:
http://www.yorkietalk.com/forums/showthread.php?t=66059 and would recommend the same to you.
Right now you need to teach her the world is a good place, and she can trust you. Lots of positive reinforcement, and gradually introducing her to new situations and people. I would allow her to be near you constantly, and take her nap times to play with your other dog. She doesn't need to be on your lap all the time, either, so your other dog can do that as well.
Do you have any background on her as far as where she came from before the rescue? Knowing a history can help tailor the training to the dog, as well.