You gotta be careful with the use of the ">" sign. Some will interpret it as "greater than" instead of "to" and you'll have a bunch of angry Texans after you :)
As an instructor for both states, I'll try to be of assistance.
The nice thing about the Arizona permit is that is doesn't have an address or anything on the face of it. To change the address, call AZ DPS at (602) 256-6280 and they'll take the change of address over the phone. WHAMMO!

The resident AZ CWP magically becomes a non-resident CWP, but even if you have it in your hand during the transformation, it's unlikely you'll feel any difference (aside from possible placebo effects LOL).
The good news is that Texas will honor the AZ CWP, even after you move and get the TX DL. You should be fine unless you come across a police officer who doesn't know the law (this is a real possibility

).
The bad news is that once you get that Texas DL, your AZ CWP magically loses some of its travel mojo.

Look at the back of the AZ CWP, it says "This permit must be presented with an official form of photo identification (driver's license from any state, identification card issued pursuant to ARS 28-3165A, military ID card or passport)." Unfortunately, some states (Colorado, Florida, Michigan, New Hampshire and South Carolina, possibly others) will not recognize a permit unless it comes from your home state. You can't even hand them a completely valid AZ CWP without proving that you don't have a resident permit
The only way to restore full travel mojo would be to still be a "part-year resident" of Arizona by getting an AZ ID card. They're pretty easy to get at the DMV, but only if you have somebody willing to receive mail for you in the state.

Really, you're better off to go ahead and get a TX CHL. Not only will you get a good lesson on Texas law, but you'll have a home state permit that's for sure good in all of the states a Texas CHL is recognized in. But go ahead and keep the AZ CWP--it's good for a backup permit in the event you misplace your TX CHL, and unlike the TX CHL, your AZ CWP is good in Vegas

.