I don't know if I'm what you'd call a good writer (I only write a blog for fun, I'm no author) but I do it just like you describe. Write the text, then go back and proofread - again and again.
While writing, I concentrate on what I'm trying to say and the wording - content is king. Get all the things in that belong there and in the right order, and written properly (grammar and what have you.)
While thinking over a paragraph before writing it, I'll usually re-read previous paragraphs. I'll often find typos and "thinkos" (wrong word or similarly spelled word.) I generally correct those when I find them.
When the text is all together, I save it and then preview it.
I write in Markdown using a simple text editor. The Markdown code is converted to HTML which I preview in a browser. The text in the browser looks very different from the text in the editor - the layout is different, it uses a different font, paragraph breaks and line breaks occur at different places, the pictures are visible, etc.
That difference in appearance makes it easier to spot wrong things. When you go to look at the text you typed, you've already seen it. When you re-read it, you remember the last time you saw that passage. It looks the same as when you typed it, so it "looks right" even if there are spelling, grammar, or punctuation errors in it. The difference in appearance between the typed text and the preview text is enough to break that connection between the memory of what you wrote and what you are reading.
I re-read the preview a couple of times, looking at flow and content (and catching the occasional typo or punctuation error.) I improve things in the Markdown file, regenerate the HTML, then re-read the preview.
Once it reads well, I make a last pass (or two) to look for spelling, wording, and punctuation errors.
One thing I find that really helps to spot mistakes is to read the text backwards. Start at the bottom and read the last sentence of the last paragraph, then the second to last sentence of the last paragraph, and so on. The idea is to make the text not "look like" what you typed so that your memory of what you intended to type doesn't interfere with seeing what you really typed.
The basis of the way I do proofreading is an experience from technical school in the US Air Force.
I learned drafting in the USAF at a time when it was done on a drafting table with pencil, pen, straight edge and compass.
Every assignment ended the same way: Run a blueprint of the finished drawing, and turn in the drawing and the blueprint for grading.
It never failed: You would spot an error in the blueprint that you missed while reviewing the drawing. We weren't allowed to correct the drawing and reprint. Once you'd made the blueprint, you were done and had to turn in the assignment.
The difference in appearance between the drawing (black ink on mylar) and the copy (blue on paper) tricked your mind just enough that you would see things that were otherwise blotted out by the tendency to see (or read) what you meant to draw (or write.)
That experience is always in the back of my mind when making/writing/doing things. A slight change in appearance often makes it easier to find mistakes.
By the way, I don't use a spellchecker or a grammar checker. Using them while writing interferes with getting "words on paper" - it distracts you from the task of formulating your text. I could run a separate checker after I'm done, but I don't usually bother.
The same goes for formatting. Formatting while writing invites you play around with things that have nothing to do with the content of your text.
Markdown is great in that sense. It has very little in the way of formatting. It has (mostly) semantic formatting - what things are rather than what they should look like. You tell Markdown that a particular piece of text is a heading rather than "font size 20, bold, Helvetica." I don't really care (while writing the text) what it will look like. I flag titles and headings and what not, then let Markdown (and the HTML converter) decide what they should look like.
If you are writing text for print, you might consider Markdown or Latex instead of using Word or other "what you see is what you get" editors. By separating the text from the appearance, you can concentrate more on what you are saying - and it makes proofreading easier since you get a fresh view of the text in its final form.