The dog is yours. Stop communicating with its former owner and do not allow her to have the dog again for any length of time. If she wants the dog back tell her that if she wants the dog she is going to have going to have to sue you and if she sues you, you will counter-sue for all your expenses in caring for the dog (food, vet, etc) as well as a daily boarding fee for the time the dog has been with you ($25 - 35 per day). Tell her she is not to contact you again unless it is through an attorney and if she does you will go to the police and file harassment charges.
If we do not have a signed fee agreement I am not your attorney and this is not legal advice.
You use the word "adoption" which if completely officially would signify a relinquishment of all rights to the dog. Were adoption appers signed or was this merely "you keep the dog?"
If I wanted to get Barb out of my life I would consider sending certified letter advising her that she gave up the dog and has no rights to it anymore, which could be used / reported to police if she attempts to take the dog back. To make sure the letter has all the necessary legal verbage, I would consult a local attorney.
Sounds like there are a lot of fine details that need to be worked through in your case. The transfer of the veterinary could be interpreted as an act showing transfer of ownership, but it is not ultimate proof of ownership. She could file a lawsuit seeking the return of the dog. You could counterclaim for the expenses that you have paid for the dog.
If she files a claim against you, you will need to seek the advice of an attorney.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.