Not sure about availability from gifts of a lifetime, but we love our GOH (Guest of Honor) badges. We first purchased some at DL, but when we were at WDW last time, dh lost his, and we replaced it at a store in MGM/Studios. Can't remember the name of the store, but it was on the right side from the entrance as you walked down the street. Funny though, when we asked everyone from the resort cm's to cm's at the MK, etc. everyone had a different story about where and if we could buy them. Most reliably I heard the Christmas store at DD and the shop I found at MGM. Don't worry about getting them ahead of time.
Oh, and I have never seen one with your state on it, though our original ones had
Disneyland on them, so when dh lost his, we all got new ones that say Walt Disney World. Didn't know if it was bad form to wear ones from DL at WDW, LOL. They also sometimes have had different colors available, with different characters on them as well. I believe it is just CMs who have their homestate on them. (just as you cannot buy a white one)
Only you will know if you are comfortable having your child wear a nametag. We find the benefits and joy outweigh any fear of bad possible scenarios.