I feel conflicted between family and a new guy

You are going to have to choose which matters more to you. If you go against your family's wishes, they will get over it and they will probably come to respect you. If they choose for your personal decisions to keep them distant from you then it won't be your fault. It will be theirs. You can't make everyone happy. You're not doing anything wrong by making your own choice. They don't get to decide who you like or don't like. It is a very personal decision. You're not hurting anyone by making your own choices. They will have to learn they can't control you. It will probably help them in the long run rather than hurt them and it's not okay for them to hurt you. How is that fair? They are not going to be in your relationship. They are not going to be there every day with the person you wind up choosing. Think about it. It makes no sense. So go with what you want. If this guy really means something to you, ignore what they say.

In this situation it seems like your family will have to be the ones who compromise. Your happiness should be their first concern, as it should be yours. If he makes you feel happy and your family cares about your happiness, they will have to come to terms with it at some point.

I'm sorry to say that someone is going to get hurt. It's surprising that your family is trying so hard to get you away from this guy? What is it that they don't approve of him or the relationship? Because that should be a red flag if they want you to see someone else. The question is, do you want to avoid creating tension in your relationship with your family, or do you want to go against their wishes and continue to see this guy (not that they won't eventually get over being upset)?
I honestly can't tell you what to do. You have to think about it very seriously and decide for yourself. You're old enough to make your own decisions, so take the opportunity and make the most out of it. Because in the end, you're going to choose, no matter what anyone tells you, and either way, there will be consequences. Good luck! :)

Thank you rskiier13, SilentRain, and Vivid Melody for your advice. I realize the larger problem can't be resolved right now, but the issue with all these blind dates is more immediate. They think I would be making a mistake to "burn the bridges" and be exclusive to someone at my age when its a prime time for exploring my options. Just something feels wrong about going on blind dates behind his back. And I tell them I don't see the point because I know I can't go on a second date with anyone because I can't have two relationships. And they tell me it's just good experience to date and meet new people.