Is it normal for the Mesquite tree to have really dark places on the bark and drip sap?
It sounds like the tree has slime flux. The jury is out on this disease in terms of how lethal it is. Some people feel that Mesquite can recover from it fine after being infected while others feel that it is potentially dangerous to the long term health of the tree.
There is no known "cure" for slime flux. It is something the tree has to battle itself. But, proper water and nutrients will go a long way to helping it recover.