Yeah, you are slowly falling into the friend zone. You begged her to give you a chance, and it seems like she is giving you one out of pity. She is a nice girl and won't do anything to hurt you, but if she doesn't like you, then there is nothing you can do about it. Just want to warn you.

As for a date, avoid the movie (Pirates 3 wasn't that good). Just one afternoon, head out to a coffee shop and just chill and talk with her. Ig the weather is nice, take a walk in the park. Just keep it nice and simple and most importantly FUN.