If you need to re-learn the reflexes to keep the bike balanced, then slam the seat down and remove the pedals, then walk the bike around. That's the way the bike's predecessor, the Draisienne, was propelled, and it's still the best way to learn. When you get to the point that you're only touching the ground to propel yourself, then gliding, you'll be ready for the pedals to go back on.
It's not surprising that a 'Cadillac' guy would have a big-box-store bike. The general attitude among non-cyclists is that no bike should cost more than $150.