You've cranked the thermostat down, your AC is running constantly, and the back bedroom still feels miserable. That's a really common situation in Tucson, and most of the time the attic is where the problem starts.
Here in southern Arizona, attic temperatures in July can climb well above 150 degrees. When insulation is thin, old, or just not the right type, that heat pushes straight down through your ceiling into your living space. Your air conditioner has to fight it all day and into the night. The result is high electric bills and rooms that never quite cool off no matter what you set the thermostat to.
The good news is that this is a solvable problem. Understanding what's actually happening in your attic, and what your options are, makes it a lot easier to decide what to do next.