You will never wobble to the right or left. Get all that heavy cargo loaded and head out. There’s no trick to stop that and there’s no cure for stupid. Or, at least, you won’t fall down unless you do something stupid like walk into too-deep water or off a cliff edge. I started to experiment with this and it turns out that there’s a very simple way to carry a lot of cargo and never fall down. In any case, this is where things get a little. Sometimes you’ll start to fall or go too fast down a hill and you can press R2 and L2 at the same time and this will ground you, hopefully preventing a fall which will in turn damage your cargo.
As you’d imagine, this all becomes much more difficult when clambering up or down steep hills or navigating infuriatingly rocky terrain.