"I don't know how to convince myself that I don't need to wash my hands, or that I really did lock the door.."

If you observe the process of becoming convinced that you need to wash your hands, and the process of becoming uncertain whether you locked the door or not - perceive it, sense it, watch it somatically in real time - you unlearn your conviction, certainty, or "nagging doubt". The certainty that you need to wash your hands is the end point of a process happening in psychosomatic space; subjective space. That process takes time; it's not instantaneous even though it might seem to be. Because the process takes time it can be felt/sensed/perceived. That is not really any different than feeling that you are hungry for instance, it just takes more subtle attention because we're not used to observing processes that happen so quickly.

But If you try to convince yourself on the heels of a compulsion, after the fact of it, you will tend to suppress it and its causes, which makes it harder to perceive how it comes about in the first place. Sense the beginning glimmer of compulsion in real time and you unlearn it.