I have another idea for you.
How about your auto box is switching into 'get you home mode'. I think this will leave your car in a high gear, whatever speed your going. So, at low speeds you have no power at all, but as your speed picks up to match the 'get you hame' gear setting, eveything appears ok again.
Think of it like driving aroung in a manual (stick shift, I think you would call it) stuck in 5th gear. Pretty lousy at pulling off, but once you got up to about 50, everything would seem normal.
Switching the car off would give your car the chance to reset (as you said).
I once had this exact problem on a GM. Mechanic reckoned it was a stray signal from a spark plug, that corrupted the data in the auto box controller, causing the controller to flip into emergency mode. Changed the plugs and it was fine from then on.
This is just a guess, I know sod all about them, but it seems to fit your description.
Good luck with it