I'd say so. Most OEM Ford oil pressure sending units don't actually measure the pressure, its actually just more like an ON/OFF switch which tells you whether or not it has oil pressure, not how much. They're pretty simple to replace, just unscrew it from the side of the block with a wrench and screw in another. Wiring is pretty simple and straight forward.
Other possible thoughts could be a bad ground for the gauge, or a bad connection but if its kicking in once its warm then it'd rule that out. LR