I've never used a spellcheck, I use a dictionary. That might help you with phonetics, as it's 'spelled out' as it's spoken. (My grammar's not that brilliant, either!)
I think both spelling and logic are equally important here...it doesn't matter if you're an engineer, philanthropist, school teacher or go-go dancer, whatever, if you want to write for others to see and hear, you'll want to 'get it right'.
Some people spell things badly deliberately I think, kind of like 'you can't tell me how to spell', but I think bad spelling detracts from the overall read. Of course, there are colloquial spellings that can add a more authentic voice, too, as well as illogical writes that are fantastic, but my advice to myself has always been 'don't break it until you first know how it works.'
For presenting an argument, the less your opponent has to ridicule you with, the better.