The potential problem isn't with the headset, it's with the steering. All that weight (a couple of gallons of milk weigh about 17 pounds) on the bars will have a definite effect on steering response. the general guideline for touring riders is to have no more than 5 pounds in a handlebar bag and less if possible.
As recommended, a rear rack is a better place to carry heavy things and there are even rear racks specifically designed to hold grocery bags.