I have never had any issues stopping at rest stops between AZ and MT, except of Idaho. I had been up all day and then on the road for about 10 hours, heading back for an emergency, when I stopped to get a little sleep in Idaho. I had been there for about 1-1/2 hours when an Idaho State Trooper ran me out at about 3:30 am, and accused me of being there over night. I guess they would rather have sleepy driver that had been up for close to 24 hours than have someone rest at a rest stop.
I just drove to Utah and stopped there with no issues.
There is a big difference between resting and camping and it Idaho they can't tell the difference.