do you still pay the rescue to take the dog?
I'm not sure what you mean here? Are you asking if someone pays a rescue to take the dog, or the rescue pays the person who got it from the shelter?
Most rescues do not require someone to pay them to take their dog, although donations are always welcome and some owners have done that. We will normally not argue with a shelter if they want to charge us for getting a dog from them, (most wave the fee for rescues) but if someone does it on their own, then it is their own expense, as we don't want to create a 'market' for people who might try to make a profit. Rescues normally do not buy dogs (some do) but most do not. IF you arrange ahead for a rescue to take a dog before you run and pick it up, generally that rescue will cover any fee from the shelter as it is known that it is coming into that rescue program.
I hope that answers your question?