I find that students routinely divide instead of multiplying, especially if they are asked to solve for something and then calculate the result. For example, $f=ma$ gets transformed to $a=fm$ instead of $f/m$, yielding a large error unless $m$ happens to be near 1kg. Checking units is good way to check for incorrect formulae, but I've found that students who mix up division and multiplication are also uncomfortable with units. The use of calculators cannot explain lack of understanding of units, so I think the problem is more basic.
I've known school teachers, and am somewhat familiar with curriculum. Both are basically fine, and neither explains the problems I see with students in university.
It seems to me that the root problem is that students are never told, in serious terms, that getting the right answers is important. The way we signal importance is with grades. Students who have received passing grades for incorrect work through many years of school can hardly be expected to show attention to detail in university.
America (et al.) may be throwing away a generation of talent, by pursuing what seems to be a policy of unconditional approval. Luckily, there is some hope: my foreign-trained students hand in correct work, with appropriate digits and units. Quite often, they also show how they have checked the work. I don't think it's because their first language was French, or German, or Mandarin; I think it is because they went to schools where learning was still the focus, as opposed to the congratulation for simply being, not accomplishing.