Store Bought Bottled Steak Sauce in Phoenix
I know that when I look for something specialized like that, I usually check online. Be specific in your search expression, and use the Google Shopping search. For example, I typed in "mesquite steak sauce" and "chipotle steak sauce", and both turned up some interesting results. In town, you might stumble on something at AJ's, for example, but it would be pretty hit and miss. That's all I've got, sorry.
We always get HP (Houses of Parliament) steak sauce, and vastly prefer it to A1. You can usually find it at Fresh and Easy or Cost Plus World Market.
Try making a grilled cheese with a nice sourdough rye bread and emmenthaler cheese, and dip it in the HP. Sounds strange, I know, but it's so good!