The turn signals, front or rear, are not lit as running lights unless an accessory module has been installed...Double Play for front and Triple Play for rear, as I recall. Even then, I'm not sure the front would be lit. If they are on as running lights, and you do not have a module buried in the bodywork somewhere, you have a short circuit. This could be in the wiring, a connector, or theoretically could also exist in the cluster itself. The fact that the signals work (at least in part) at first, then quit flashing when warmed up, points to an electronic origin...either in an accessory module or the cluster.
Don't be fooled into thinking the computer diagnostics can tell you everything and anything you need to know. This is not something that is typically monitored. Many techs today think the diagnostic computer is a god, but it still can't find things like a bad plug wire, plugged fuel filter, bad valve, or a host other mechanical or electrical ills. Modern vehicles are sophisticated, and need diagnostic systems to tell the tech where they are "feeling bad", but they are also mechanical and electrical, and in those cases a good old-fashioned mechanic is the answer, not a black box.