It sounds like this girl likes you, and you should step things up and take it to the next level. Try holding hands and linking arms as well as taking her out on real dates, just the two of you.
Don’t worry that you’re instigating the texts and she’s not. She’s waiting for you to show your interest, and after only 1 month, she’s not going to start texting you when she isn’t sure how you feel about her. So ask her out, and see if there’s any chemistry and compatibility!
Good luck! 🙂