It seems to me that you have every right to take the family who signs and pays first. However, you should definitely take the family that YOU want to sign on because it is YOUR business.

My opinion is that the space is open until somebody signs on the dotted line, then the people who didn't jump on it get a 'thanks but the space is filled' email. You don't have to feel the least bit guilty about who interviewed first, but rather who is intelligent enough to see how valuable you are.