Hello: At thirteen years, the anode needs replacing. I wouldn’t be surprised if your tank has two. Looking in an element port will show you where anodes are as well as show you the condition of the dip tube.
I usually set heater thermostats to the same 130* top and bottom. This is hot enough to slow bacterial growth and low enough to take a long moment to scald, allowing time to move out of the hot water. Manufacturers are tired of being sued for scalds, so they suggest 120*. This is only because folks haven’t tried sueing manufacturers for Legionnaires infections … that I know of 😕