Starting Problem - Automatic Transmission Car

I turn the ignition key and nothing happens. At first I suspect a flat battery. No it wasnt that. Then I suspect the ignition cut out switch. That seems to be working OK because when I push the shifter fully forward I hear a click and the car starts. Next I notice that sometimes after parking on the driveway the shifter is in "Park" but the car starts to roll backwards due to the incline.

It seems like the shifter linkage needs an adjustment but folks tell me there is no adjuster. Someone mentions a shifter pin and wear or the need for lubrication.

Has anyone else had this problem? Anyone know how to fix it?

No I haven't tried a diagnostic tool. Isn't that intended for engine related diagnostics? Correct me if I am wrong, but I can't see how it could detect and report anything related to my current problem. The engine and transmission perform beautifully once the car starts and is driven.