Arizona recognizes all permits from all states.
You can carry open or discreetly, with or without your CHL in most places (there are few exceptions).
However, unlike Texas, if you carry where booze is sold for consumption (there is no requirement about food being served), you must have your CHL, you must carry discreetly (no open carry), you cannot consume alcohol, and the places you are entering cannot be posted "no firearms."
Others on this forum will chime in to help and correct anything I may have stated incorrectly.
When you get back to Texas, please support the Lone Star Citizens Defense League. They're trying to legalize open carry in Texas.