there are a few things possibly going on... as far as the dies when you stop- make sure you have a good taillight and brakelight filaments cause vespa's have a grounded brake wiring circuit that dies if the brakelight filament in the bulb is burnt out when you pull the levers
As far as when the rear wheel engages, there is a button and lever on the rear transmuission case....if you can start the bike it sounds like the button is extended out and in "motorized mode" right? i think you need to post pix and a more thorough description re that part
oh oh oh and thats a 1968? definitely post pics! is it a 17" rims or 19" ? springer post seat or rigid square post seat? rigid front forks and bicycle brakes? springer front forks?