Are you sure the running lights are actually coming on and not just the indicator. I had a similar problem when the element in one of the bulbs shorted to the other element.
-2001 Tahoe original 5.3 4x4 only 430k
-A Metric Butt ton of three wheelers
-1985 V65 Magna
-1983 Nighthawk 650
-1969 F250 Highboy built 360