Do t stat first. then diagnose fan clutch. Flat strap bolted to fan and pulley to lock it will tell you if you here it turning on, My truck runs hot when loaded all the time, with a 5 core rad. So dont condemn the gaskets just yet, could be but always KISS it first.
But my stat just failed and was doing exactly what yours is. , and now I think its stuck open cause its 37 here this am and the dang thing wont put out any heat.