Don’t know if it’s true for everybody, but one early warning of sleepiness behind the wheel for me is that I slowly begin to lose the ability to estimate the speed of traffic ahead of me, especially at night. Time to pull over and get out of the vehicle, walk around, get some fresh air.
I recall a period of agitated hyperactivity followed by a deep sense that everything was A-OK, a cessation of active thought, and a pleased awareness of the rhythmic engine noise. The other vehicles had become nothing more than mood lighting at that point. Your test is a good one; perceptible mental impairment of any kind is a big sign, but you have to be quick to catch it early. I was already more or less out of it for some time before the sudden snooze.