"Dropouts" are just the U shaped slots that the front and rear axles fit into. And by connection the bit of metal that forms the U slots.

Another issue that could cause what you're feeling would be that the headset (the bearngs that hold the fork in the frame's head tube and that allow you to turn the bars, fork and front wheel smoothly) may be cracked or may be worn to where there's divots worn in the cup or cones of the bearings. Rusting or riding for a long time with a loose head set preload can wear notches into the bearing races that makes it hard for the bike's steering to smoothly find its center. As a result it can end up wandering like you're describing. Those of us that know what to feel for could swing the steering side to side and know right away if the bearings are notched or not. In your case since you're not that up on your bike stuff you may want to take it back to the shop and have them check it out. Or another bike buddy that knows their stuff.

It takes very little bending to alter how the fork steers. So to the untrained naked eye of yours it may not look bent. But it may well be bent back far enough that the bike will not ride in a stable manner and that could be the reason for the wandering. The damage doesn't need to be obvious at first glance to make the fork unuseable. If the one or both legs are bent back or to the side as little as 1/4 inch this will make a very noticable difference in how the bike rides. A half inch would render the forks all but unridable.