I have both adopted and given away several timeshares, and what I would do next it to contact a reputable closing company - they will handle the whole thing for you for around $150. Since you are paying for the transfer, it's traditional for you to choose the title transfer company.
This is who I use - they are Tuggers, and they are efficient, and inexpensive:
Lisa Short and Mary Pless
You and the new owner will both provide your info. to LT Transfers, and they will do everything else.
*So I'd contact the person adopting your timeshare and say: DeniseM on TUG recommends using LT Transfers, so that's who I'd like to use to transfer the title to you, and I will pay all the expenses.