The use of "sovereignty" relating to tribes shouldn't be construed as independence, more like a kind of autonomy. Tribal members are US citizens and subject to all federal laws, and receive a lot of federal money each year. Sovereignty often means taking over federal programs like policing or health care delivery: taking the money but running the program tribally, this kind of sovereignty is also called self-determination.
Sometimes tribal members are exempt from state law depending upon where they are in the state, and which state. Some tribal members automatically are granted dual US/Canadian citizenship, I'm thinking members of St Regis located in NY and ONT, and the Lake Huron First Nations just east of Sault Ste Marie ONT. Some members pay state income tax and others don't, depending upon the state, the location in the state, and the employer. It's a mixed bag of different rights and situations depending upon the tribe and the state. Tribal courts may have jurisdiction over non-tribal members on their trust land depending upon the situation, but often they don't. Tribal courts may have jurisdiction over members of members of other tribes on their land. There are instances where a non-tribal member can hit a person on the face on trust land and there's no legal jurisdiction: neither the county nor the tribe has jurisdiction to prosecute. However federal courts have ruled that if the assault is a domestic violence situation then tribal courts do have jurisdiction.
Adding that there's a rogue Navajo tribe called Ramah that split off from the main tribe, and vast off-reservation Navajo trust lands in New Mexico outside the main reservation and not included on maps. The "Breaking Bad" RV meth cooking scenes were filmed on off-reservation Navajo land just outside Albuquerque.
I always lumped Monument Valley in with Route 66, places on the European bucket list driven by cinema and television but of less interest to Americans.
I have visited 15 of the places on the list.