I live in Arizona. Which city should I use to set my global atomic watch to sync.

The instructions say it can take a while depending on where you live..

https://www.google.com/search?q=On+Time+Talking+Atomic+Watch

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Atomic does not mean automatic. It synchronizes itself on a time signal coming from an Atomic Clock in Colorado. You really need to read the manual and keep it handy. Frankly I cannot set my Casio without its manual. Manual are available for download at casio world global site.

Your watch should be set to LA (LAX) or Denver (DEN).

It sounds like the time zone may be incorrect; check your manual to see if this can be the case. Also, these devices may wait until some time between midnight and 3 AM to synchronize with the 'atomic' standard. Make sure you have the watch near a window as this may aid reception depending on where you live.

The manual mode rarely works, because you have to be awake to do it and most of the time the best signals are from 1am to 3am in the morning.

You need to manually adjust or set the hands to the home position as they are not where the watch thinks they should be. This can happen when you bring your watch close to a strong magnetic field. Look in the manual to do this. Afterwards your watch should set to the correct time.

When the logo is not shown in the bottom corner, the watch hasn't got a signal yet. put your watch on a window sill, and hold the Forward buton until the watch says GET! Form there, the watch will either display ERROR, if it couldn't find a signal, or will have the latest and most accurate time. The watch is also probably set up for auto detect, which is best. To do auto detect, before going to bed, the watch will try to get a signal from 12 midnight, to 5 in the morning.

You don't need to set the watch back an hour.  You just need to turn off the automatic DST adjustment.
Hold ADJUST for several seconds.  Then, when DEN starts flashing (or whatever timezone you are set to), hit MODE The display will change to DST flashing at the top and on or auto flashing at the bottom. Click TIME/MEMO or RECEIVE to change on->auto->off.  Once off is flashing at the bottom, hit ADJUST twice and you will be out of DST mode and everything should be fine.