That's funny-- my family used to think that my dad was something like 1/4 Cherokee. (It got complicated, because there was a little Native American on his mother's side, too, so I don't know how the fractions really worked out.) However, some digging turned up the fact that going back a ways, he was not Cherokee, but Comanche. It happened that there were so few Comanche left in the area back in the day that they were put on a reservation with the Cherokee.

As far as it always being Cherokee, the Cherokee and Navajo are by far the two most populous tribes in the US. The Cherokee being largely settled in the southeastern US, it would make sense where you are for people to be mostly descended from the Cherokee. (Though as I say, there are other less populous Native American tribes that have been lumped in with the Cherokee over the years.)

(Completely off-topic, lol)