By “running lights” I assume you mean the parking lights.
On my 06 Monterey, only a year newer and in most regards little different from yours, the top left relay in the bay, K1, is parking lights as Steve remarked, as is circuit breaker CB7. Of the 2 vertical buss’ directly above the solenoid, CB7 is the 7th breaker down on the left one. Thermal breakers usually reset and rarely fail, but connections can loosen or corrode causing resistance.
Also check the vertical common terminal buss, TB1, for the connection integrity of the top wire(s) and nut. That should be the one for the parking lights; of the two far right buss’ in the bay, TB1 is the left one.
Joel