Well, blackberries grow wild in the Pacific Northwest. If you take that as a starting point, I'd say you're really pushing the zone trying to grow them in Southeast Arizona. So you're going to have to help them out a lot. First, are the pots big enough? These get to be quite large plants. Insulating the pots so the roots of the plant stay as cool as possible will help.
Try placing them where they will get only morning sun, and will be shaded in the afternoon so they don't get too hot. Water them thoroughly in the mornings and again if you can after they are in the shade in the afternoons. I agree with Alyssa, I don't think you can overwater these plants where you are. Spraying the leaves with water wouldn't hurt either. They are native to a place with much cooler temperatures than AZ and lots of rain. If you want them to grow and be happy, you're going to have to try and make them feel at home.
