Hi, Allen! I feel your pain! I am at the end of the "teenagerous" years with my nearly 19 year old daughter, and I have a nine year old who is getting a jump start. I have to say, if you can adapt to your budding teenager rather than expecting her to adapt to your desire for your little princess, things will run much more smoothly!
I learned with my oldest daughter that the best thing for me to do was try to put her in charge - let her be the lead in terms of deciding where she wanted to eat, what attractions were important to her. My daughter had the biggest crush for the longest time on Johnny Depp as Captain Jack Sparrow, and I recall one trip where she had us ride Pirates of the Caribbean three times in a row!
Having your daughter participate in the planning also allows for some negotiation. If she chooses where to eat dinner one day, you might choose lunch the next day - and pick someplace Princess-y!
You may be pleasantly surprised, as I know my young teenager often would still grab her Dad's hand to drag him in the direction she wanted, and it really warms my heart to this day to see her walk up to her Dad and want to hold his hand as they walk through the park. At 19, I expected her to long ago tire of us!
I'm sending you a bunch of virtual pixie dust to help get you through not only this vacation, but the entire teenage years. The thing that's kept me going is the fact that there are so many grown-ups in the world - so obviously, Moms, Dads, and teens alike manage to make it through just fine!
Have a great vacation!