I'm no expert on desert plants, but I will throw in my 2-cents worth. The tree is a palo verde. Not sure what type. It appears from the first photo that a large chunk of the tree is dead. The bark should stay green, even in the winter. The last photo looks really bad. When the bark can be peeled off easily and there is mold growing underneath, it is a sign that that branch is dead. You might be able to cut out the dead branches and see what you have left. But, it might just be that the tree is old and is dying.
As far as the irrigation goes, I wouldn't keep a palo verde on drip irrigation. These are highly drought tolerant trees. After they are established, they can get by on whatever mother-nature provides. They will look a little more lush with irrigation, but it isn't essential. However, if you are going to irrigate, it is better to deep-irrigate less frequently, rather than give the tree a half-gallon twice a day.
I hope this helps.

Stop watering it so much! The amount you are watering it would keep a cottonwood alive much less a desert tree such as this. The rot you are seeing is VERY indicative of this.

Please study up on the requirements of this native plant. You are killing it with kindness, unfortunately. Good luck.