First and foremost I don't know anything about writing outside from what one had in school. I am a software developer but I think there is actually an overlap here and I could offer some perspective.
As a software developer, in my team we have code review. Where we basically look at code from other people in a tool and leave comments. These can be obvious mistakes, things that could be done better, general questions why something was done a certain way and so on. The end goal is to improve the quality and train people.
After doing this for a while one learns to catch the common mistakes and kinda develops a feeling where something might be off. What helped me most in the beginning was seeing what other people commented on the same piece of code I also looked at. Through this I was to learn to see things in a new perspective, got to learn what other people find important and how they look at the work.
So if you can review some writing of someone and also getting to see what other people commented about it, it could really help. It can train you to think more like other people when reviewing, which then improves your skill to self edit dramatically. Since you can kinda "simulate" an outside perspective. But even without it just reviewing others people writing will still improve your skill to find mistakes. Since either way one develops this skill to spot the mistakes or just this sense of "something is off".
Also a big part of being able to review myself is that the review tool present the code in a very different way, font, color, spacing .... This forces me to read it as it was something new and I don't fall into the habit of glossing other parts subconsciously. It creates some sort of distance between what I see when writing and what I see when I see when reviewing.
The last part is basically what @RiverTam suggested. Changing how to present your writing at least should force you to read it more carefully. This should help you to find a lot of basic mistakes.