"I do not want to visit you today."
Translation:Nie chcę cię dziś odwiedzić.
On its own - yes, but I think that then English should also have "today" at the beginning.
"Dziś nie chcę cię odwiedzić" sounds to me as you contrasted 'today' and other days. Like, normally almost every day I feel like I want to visit you, but somehow today is different.
True, but that negation applies to the rest of the sentence anyway. It's Genitive. If the sentence had 'her' instead of 'you', it would be more visible: "Chcę ją dziś odwiedzić" (I want to visit her today) vs "Nie chcę jej dziś odwiedzić" (I do not want to visit her today).