Almost any bank that issues credit cards will issue a "supplemental" card, that, in your case, would be credit card with your name on it, but as an extra card on your Dad's account. That will make paying for things a lot easier.
If, as was stated above, you look clean cut and responsible, use those rarest of words amongst young folks, "ma'am" and "sir" you likely won't have a problem.
If anyone starts into the 'you're too young routine' keep your cool and your manners, but remind them you're just looking for a SAFE place to spend the night. We all forget the little stuff sometimes.
Enjoy the trip!