Hi Harry and welcome to the forum. If by "historical sites" you mean old battle grounds and such, I don't know of any that are open to detecting. But New Braunfels is an old town and there are many possibilities for you to consider. You will need to check your city ordinances but Landa Park should offer good hunting if you can pick your way through the trash as will any of the parks along the river. Even better are the old private residences. Yes, you have to ask for permission but it's worth it. If the house is currently occupied just knock on the door and explain who you are and what you would like to do. Be sure to say that you will not be digging big holes and that you will leave the yard in at least as good a shape as it in now. It helps to practice your spiel ahead of time. If the house is vacant you will need to find the current owner. I live in Bastrop County and use the County Appraisal District GIS map to do this. Once I know the name and address of the owner I use the internet "white pages" to try for a match. Then I call and make my pitch over the phone. I've found that I get permission over the phone about as often as I do in person. If I'm refused permission I thank them for their time and close by saying "Give me a call if you ever decide you want to know what's hiding in your yard". If I'm talking in person I give them a card with my phone number on it. This often plants a seed in the owner's mind and they start to wonder about what might be hidden out there.
Another of my favorite places to hunt is older churches. Kids are given change to put into the collection plate. Then they start running around playing and not all of those coins make it inside the church. Women tend to wear their good jewelry to church and some of it gets lost. I approach the minister for permission and offer to split my finds. I also tell him that should I find jewelry I will notify him so that he can announce it to the congregation and give them my phone number. If a church member calls and describes the item I've found I return it to them. It is great PR both for you and for metal detecting in general. As a side note, I've never had a minister accept half the change I found especially if I show it to them before I clean it up.
Let me know if you ever get over Bastrop way.
Storm