I love this topic and in my short chance to get on the internet I wanted to be a part of it. So, to me, there are two routes and a middle ground to pursue that are entirely up to the writer - because the reader will interpret them very differently.
The first route is to do exactly as you have - state that the character speaks that way, establish it early, and mention it now and then to keep the reader immersed and sounding it out in their head. The pros to this are that you can still write fluidly without having to sit and think overly long about how to spell it out. However, using this method you should still try to throw in the vernacular and slang of whichever dialect you're planning on using. The cons here are that it can seem, to the reader, as a lazy way to circumvent the challenge facing the writer, no matter how many dialects or accents are out there. And that's unfortunate, but in some ways it's true. Especially if we're not just talking about an American from the Deep South, but rather a Scot - where they actually spell words in social media with their accent (it baffles me and is almost unintelligible; one of the most amazing things linguistically I've seen - I love it).
That brings me to the second option, which is to spell and write as though you're speaking it - using phonetics in dialogue because that beautiful little margin of space between two quotation marks is permitted to be a veritable wasteland of grammatical anarchy and character reflection. Pros to this method are that, in my opinion, it's more immersive to an extent. If you can do it right, though far more difficult, it helps to really feel a character out just right. The downside is the difficulty, the time it can take to think and write it out and research it, and the fact that whether done right or not can make a conversation seem jagged and tedious to the reader.
So, when I'm reading at least (I struggle with how I write the characters myself, to this day, but try to lean on the second option moreso), I prefer that perfect middle spot. The place where a writer and speak through the character without sacrificing the message and comprehensibility of the words just for a gimmick. Especially when the world is primarily one accent - if everybody has a drawl, then perhaps stick to the vernacular and keep the phonetics to a minimum. However, if you're in... say, England, and all of the characters have a posh, well-educated, measured speech until some Southerner from America shows up and starts spouting with a twang - even if educated - you'll want to use that chance to show the dichotomy of cultures.
Just food for thought - these might be ramblings, I just got off a really long shift and have had only sporadic days off lately, but I really wanted to weasel in on this discussion.