I doubt that’s what the problem is. But no harm done trying it. Might work. I expect however it is either (as posted above) that relay is sticking, or my guess, is (again as posted above) a faulty ignition switch. One idea, try driving the car with just a single key in the ignition switch, nothing else dangling. If that helps, points to a faulty ignition switch. Such a thing isn’t uncommon on a 16 year old vehicle. The ignition switch on my Corolla went a bit faulty at about that age.