Here's my experience: My thornless blackberry plants LOVE water, even more than the raspberries. They are in the ground though, and I water them at night, when needed, to let the water soak in overnight. Essentially, I don't let the blackberry bushes dry out.
I used to live in AZ, and I'm thinking yours are getting too much sun and not enough water. They may be heat tolerant but I wouldn't push it with full sun all day. Maybe behind a pillar, or under a tree, or porch area would be best for the pots.
Would you be able to upload a photo of the plant you're worried about? That way we could confirm if my thought is correct.
