If you stab the brown a bit with a screwdriver does it sink in?

I think all Palo Verde variants have a problem with branches dying for no reason at all, sometimes a bunch at once but usually one big 'un at a time. How old is this tree?

These trees get very erratic water from rains and grow in little spurts when they do get some. I think it's a mistake to water a mature Palo Verde because if they grow "too well" they grow weak and punky and break easily. Watering also can exacerbate root rot problems in desert trees. My suggestion would follow your own hunches: let it dry out and prune to relieve dead branch weight.