Another issue that I did not see specifically mentioned is how smoothly you pedal. I'm sure someone could add a lot more technical information here but in layman's terms, if you have a nice round pedaling stroke the bike and load are generally more stable. The smooth pedaling combined with good core body strength go a long ways towards preventing the development of dynamic sway, i.e., wobbling while pedaling. Clipless pedals, or toe clips and straps, let you pull and push through the rotation. Some slight pull at all times helps smooth things out and a lot of pull helps you power up hills or accelerate.
It took quite a while for my mid-teen son to develop a smooth pedaling stroke as both technique and strength are involved in doing so. To help him understand what I was talking about I told him to watch how well his rear wheel tracked behind the front (relatively dry roads with puddles or cinder rail trails help here). When he saw how much his front wheel was constantly going from side to side as compared his rear (and compared to my tracks) he understood. After a couple thousand miles of loaded touring over two summers, growing in height and gaining a lot of core body strength during that time, he pedals much more smoothly now and it shows in his tracks.
I noticed that it also takes a bit of time at the beginning of a tour for my body to automatically and seamlessly make the core body muscular adjustments needed to keep the bike steady. I'm talking about a few hours at most... By the second day the muscle memory is restored and there are no more wobbles.