A confused mind and an unsteady body share this word’s history.
Dizzy continues Old English dysig through Middle English disy. The older adjective could concern foolishness or confusion before the familiar bodily feeling became central.
Source
Merriam-Webster: dizzySelected historical forms, oldest first: Old English dysig → Middle English disy → English dizzy. Each form is in the ancestry rather than merely a listed cognate. Selected glosses: dysig: Foolish or confused; disy: Dizzy or foolish; dizzy: Feeling unsteady.