Hm, shouldn't you have your foot on the foot brake unless you intend to use the accelerator? Then this whole "handbrake not grabbing between gear changes" thing wouldn't be a problem.
Although now that I think about it, I think I may have a similar problem. When parking I pull the handbrake as normal, and put the car into park, but it rolls slightly before you near a click, which I believe is the parking pawl stopping the car. I do generally park on a downward decline though. Not saying that the car moves between gear shifts, but the handbrake must be weak or something.