You can use the search function to look for things people do that contribute to wordiness or other problems of clarity. If you search for occurrences of the word “was,” you can find all the places where you said “was [verb]-ing,” when “verbed” works just as well, for example. When I write something really important, and more than a couple of pages, I always do this. I search for the word “like,” and “such,” to find similes that would work better as metaphors (where I could say that x is y, rather than x is like y), or even just incorporate an image into a phrase, and lose the copula altogether (real example: once changed “until the sun was a little, glowing stub” to “until just a glowing stub of the evening was left”). I check places I may have used the subjunctive incorrectly. I check for sentences that begin with conjunctions, to make sure I really want to do that. Prescriptivists say “don’t do that” for a reason. It still can work in informal writing, but you really, really want to make sure it’s the best choice, and that the sentence is clear. Search for the word “can,” and change it to “will” if “will” works, to strengthen your sentences. Get rid of qualifiers like “sometimes,” “might,” “could,” and “sort of,” unless they are imperative. Just say your bit, and go on. The more you qualify, the less punch your verb has.
Grammar check detects run-on sentences, and other things that hurt both clarity and style.
I second the recommendation to put something away for a couple of days, and try not to think about it, then approach it as though it isn’t your own.
When I edit, I try to cut as many words out as possible. Verbosity is the enemy of clarity. Any time I used three words when I could have used two, I change it.
FWIW, people tell me I write well, and I am usually to go-to person at work when someone needs something proofread or edited. I was once on a team that had to turn in reports every week, and we passed around the job of writing them. Our supervisor told me in an evaluation she was always glad when it was my turn to write the report, because mine were “easy on the eyes.” I’ve got a degree in English, and I pretty much always got As on papers. I got a few Bs in English classes my freshman year, but pretty much all As. I’ve written movie reviews for two different local rags, and I’ve had a couple of short stories published. I’ve never been more ambitious than that, except in my daydreams, which is why I’ve never written a novel, other than the one I wrote to cheer up a friend, in which I made her the main character. So that’s where I’m coming from when I make those recommendations.
Of course, always spellcheck. Even really good writing will be brought down a notch by bad spelling.